I enjoy my road bike, but recently my children have become old enough to ride, but still very slowly. I'd like to be able to ride with them on all or mostly flat pavement, but the road bike just isn't pleasant for slow riding as it takes more effort to balance and isn't that comfortable.
What type(s) of bike is well-suited for very slow comfortable riding like this?
Any bike with a comfortable seat, upright riding position and low ratio gears.
This is pretty much the definition of a 'cruiser' style bike.
Many hybrids, less expensive mountain bikes and fatbikes would fit the requirement too.
You could consider a 3-wheeled recumbent bicycle:
Photo courtesy of Willeke
These tend to be more comfortable, and the three wheels mean balance is not an issue, so you can go slowly or stop altogether without any balance issues.
(Note that there are also two-wheeled recumbents, but they would probably not serve your purposes as well.)
You might want to consider a fat-bike. They're naturally slower, because they're meant for really rough terrain, so they have low gearing, high frame mass to be sturdy and higher rolling resistance due to the fat wheels. The fat wheels also make them pretty stable.
Plus, if you're so inclined, after you don't need it for going slow anymore, you can keep on using it to do completely different kind of biking than your existing road bike: rough off-road.
Get a "mountain bike" rather than a road bike. It's what I think of as a normal bike anyway, but they don't go as fast. It's easier to keep your balance on them too.
A cargo tricycle? Bonus is that you can save gas getting groceries with it.
Generally speaking, a bicycle with a longer wheelbase will be more stable when ridden slowly. Those designed for quick maneuvers are generally very short (criterium) wheelbase designs. But the tradeoff is, of course, that the long wheelbase makes those quick direction changes a bit more difficult.
Modern bikes tend to have straight forks and very steep head angles (where the fork attaches to the frame). Both of these shorten the wheelbase and make the bike more "responsive" aka "twitchy." Shallow angles and curved forks go the other way. Stability and comfort.
I hate to recommend anyone get a heavy bike, but there is some effect. The lighter the bicycle, the higher the center of gravity overall (i.e. you).
On the other hand, track riders can ride slowly, even stand in place. They do this with skill and the lack of a freewheel. You can even ride backwards on a track bike. These bikes are also twitchy in the extreme. But it is rider skill. With enough practice you can learn to "stand on the pedals". That is, balance the bike in place, though with a freewheel you need to also use the brakes You've probably seen people doing this when stopped at a light, balanced upright.
I have a foldable bike which is quite comfy at low speeds (especially so probably due to the low CG). With the added advantage that you can transport it easily to kid friendly bike paths.
