For most casual group rides I've ridden, there are points where the lead pack pauses to wait for dropped riders (usually at the top of climbs). When the last dropped rider has regrouped, the pack immediately starts off again. I've always wondered why this is, would not the last rider be the one most in need of rest (especially if it's me!). If time were an issue, why not just keep going leaving slower riders to go at their own pace? If concerned about keeping the group together, why not slow down?
There is a lot to unpack on group riding etiquette and waiting for slower riders. Some people ride for social reasons and some people ride mainly for fitness so it can be difficult to balance everyone's need in a large group.
The system we use that I think works quite well is we tend to have about five groups of varying ability and we stagger these groups out over our ride route. The fastest ride is always a drop ride, meaning that no one waits. These are fun but they are intense as it is full on race pace. The idea is if you can't keep up you can always drop down into a slower group that will be arriving in a few minutes. All of the slower groups have a no drop policy. This lets people filter to their ability and it seems to work quite well as everyone ends up finding their preferred group.
When you only have one or two groups, a no drop policy can cause more friction especially between those that are riding for social reasons versus those that are riding for fitness reasons. Everyone is here to ride, but some have little interest waiting around and socialize they rather do any socializing while on the bike and turning the cranks. For them the act of waiting is already considered a courtesy. They are simply looking to get moving before cooling off too much and are probably not thinking about how tired the person catching may feel (or forget because they are rested).
Another part of it is the HTFU (Harden The F%#@ Up) mentality that has long dominated road cycling culture. There is the belief that suffering is to be expected and embraced. As an extension there may also be an expectation that weaker riders should work to improve their fitness and or strategy (e.g., skipping pulls so they can stay rested for a climb) in order to stay better within the group.
As road cycling has been going more mainstream over the last decade or so there have been some growing pains over cultural aspects such as these ones.
Which ever way you slice it it can be difficult to meet everyone's needs especially as the size of groups gets bigger. The best you can do is looking for a group that best jives with what you want out of cycling. For example, I actually look for rides that drop slower riders (and I have been on the receiving and giving end over the years) because I am specifically out looking for a challenge rather than a social ride.
Depends on the ride culture and how familiar people are.
When I go riding with a group, its generally with a decent variation in riding ability and familiarity in riding in a group. Also, not everyone is familiar with routes always. So, we often split into two groups: one fast, another slow.
Slowing down the fast group is boring. The slow group motivate each other to keep going at a reasonable pace. And if someone needs to stop (medical, puncture, etc.), the slow group often does.
I think the idea is that by waiting for you at the top, and letting you rejoin, they are already providing enough help for people who aren't as strong riders. Once you have rejoined the group and are on the flats, you can benefit from the slipstream and should be able to keep up with the group. If you are getting dropped on the flats, or are so tired when you catch up after climb that you can't even remain in the group with the aerodynamic benefit of the group, then perhaps the group is just too strong and you need to find a slower group to ride with.
This is coming from somebody who very much isn't a strong rider, but also someone who doesn't have a lot of time to ride. I understand why the fast riders wouldn't want to wait up for people. We usually split into 2 groups, fast and slow on my group rides, and I'm usually working hard just to stay in that second group.
Even as a slower rider there's been times when my ride has been ruined by people coming out for their first time and clearly not being fit enough for the ride. There's people who show up for the 50 km ride who probably haven't done more than 20 km at any point. They hit the wall or literally crash from exhaustion at 30 km and then have to limp back all the way to the city.
When this happens I try to remain positive and encourage them to finish the ride, so they don't get turned off the sport entirely I make sure they get back home safely either by going slowly until we get back, or making sure they have a friend coming to pick them up on the side of the road. But in the back of my head, I just keep on thinking about how my ride kind of got ruined by having somebody tag along who clearly wasn't ready for the ride. We let people know what the ride entails before setting off, and most people should be able to assess whether they can do the ride with reasonable certainty.
On a friendly ride of mixed ability some people will be happy to get round the route intact, while others will want to race each other up hills or on straight sprints, or just find a slow pace frustrating compared to a quick ride and a wait. But they may still enjoy a friendly ride even if it's a little tame for them, and their experience is welcome).
The slower riders do expend less effort because even uphill air resistance accounts for a fair bit of energy. So while they plod up the hill the sprinters can get their breath back.
My attitudes to group activities were formed from hiking and kayaking, so I reckon that no one should be left alone at the back, and the back group should include someone with tools, map, phone, etc. but who could also catch up to the main group if necessary (e.g. me as I'm a bit of a pack rat but not too slow). Others differ on this so we choose different rides, to the extent that if the word drop is used except in the sense of not believing in the concept, I'll leave that ride to other people.
Group rides are usually described as drop rides or no-drop rides. No-drop actually has gradations within it, which I discuss below. The behavior described above is consistent with a no-drop ride executed poorly - the OP was absolutely correct that if you've dropped someone, they're going to need to catch their breath.
Some no-drop rides interpret the policy more strictly. Often they have a sweeper, so if someone really can't handle the pace, the sweeper will ride with them on the route or help them take a short-cut back to the finish.
Some no-drop rides interpret this less strictly. It's more like they won't drop you unless you are way behind the group in group riding skills or fitness. To be fair, if you are not comfortable riding in a group and you hang several lengths back but you also don't have the fitness, it will be very hard to stay with the group. Calling yourself no-drop but using this interpretation of the policy can cause friction if there's a wide spread of ability levels.
It's better to be clear about the policy than to be rely on implicit information, which new riders are obviously less likely to have. I think that if I were organizing, I would prefer to call this a drop ride, but make explicit that we will wait after climbs (and probably major descents) or other sprint points. Other drop rides are less merciful and will wait only minimally. For some really fast rides, this is just how it is, and if you are fit enough to consider one of them, you're also experienced enough to take care of yourself.1
Ride organizers can help by posting GPS files of the course. I like to save them as a cycling route on Strava, and then set my GPS to sync to Strava. Otherwise, you can upload a .fit or .gpx file to your bike computer, but this takes an extra step. Of course, newer riders are less likely to know how to upload the routes. Printed cue sheets are or were also a thing, but again, newer riders may have trouble interpreting a cue sheet, especially if they've gone off course.
As I mentioned above, if you have just had a sprint, stragglers need time to recover. Depending on the terrain and the intended pace after the sprint, some of them can just sit in to recover. Ride leaders need to consider this - most likely the ride leader in the question just failed to do so, maybe due to inexperience, maybe because it's been a fast group before. You need a plan about how to handle drops and splits (if you want the group to stay together). In my club, the people at the back call pace (repeatedly) to signal to slow down. Importantly, if you do this, people mid pack need to echo the call up to the front, and the people at the front should verbally acknowledge in addition to slowing down. Otherwise, splitting the group on the road or at the ride start, as discussed in other answers, is a reasonable strategy. Naturally, ride leaders should still assess the spread of ability levels in their ride just to be safe - it's better to be aware if someone less strong than the average might be in danger of dropping.2
Leading group rides is a skill that you have to learn. Moreover, so is being a good teammate. Ride leaders need to maintain situational awareness, and they should keep a count of riders. Other people on the ride need to help alert the ride leader(s) that someone has dropped or is in trouble, or that a gap has formed. This can include sprinting up to the front to tell them to slow down or that someone dropped off. Nobody has complete situational awareness. Moreover, your hearing is reduced while riding due to wind and other background noise. Thus, being a good group, especially on harder rides, takes a team effort.
This attitude is appropriate in the age before indoor training became widespread. There are a handful of riders with considerable fitness from indoor training who have minimal road group ride experience. Clubs who encounter them might want to think about how to include them on rides. For the question's purpose, it should suffice to wait longer at critical points.
Be aware that people can and frequently do strain to hang on, then crack massively. Often they are using anaerobic capacity (colloquially called burning a match / matches) to hang on, but after this is exhausted, they need to ride significantly below aerobic threshold to recover. From the outside, it may look like someone was hanging on but they just disappeared unexpectedly. Frequently a hill will trigger this. Alternatively, riders in difficulty may give cues, e.g. body language, facial expressions, frequently letting a gap establish if they otherwise had the skill to ride in formation.
In more wild terrain it may be extremely unfriendly to drop members behind without caring what happened to them. What if the bicycle breaks, or worse. Hence the team or part if it should pause time to time till the last straggler is seen again. Otherwise they are ar risk to be forced to turn back looking for the missing members. This is the minimum and the team may opt not to wait for longer.
Who absolutely want to ride fast, should just form the group without weak riders. But if already happened to have somebody weaker in the group, abandoning the one in the middle of mountains does not look cool.