My chain was badly worn. 1.0+ % wear. There is no visible shark toothing or extreme spacing on chainrings. Is there a chance I can get away with not replacing the chainrings, or does the amount of chain wear automatically necessitate that I replace them?
I think that one rule of thumb I saw asserted that if you replaced promptly, you could get 2-3 cassettes per chain, and 2-3 sets of chainrings per cassette. I'm not able to provide a concrete source for this right now. However, the fact is that chainrings do wear more slowly than cassettes because they have more teeth.
I think very likely that your cassette needs replacement, and I would do that preemptively. I would not replace your chainrings right now, unless they have had several prior chains that also got to a high level of wear. For smaller chainrings, I would probably lean slightly more towards preemptive replacement.
When should I replace my....
Chain? When the chain checker says your chain is over 0.5% worn.
Cassettes? When you start experiencing "skipping" in your most popular sprockets.
Chain rings? When you see very obvious "shark teeth" on your chain ring teeth.
Yes, it is true that there are more reasons why you should replace these components, but these are the "most common" in my experience. And remember: a worn chain will wear down your chain rings and cassette, but worn chain rings and cassettes will not wear down a chain.
Two main schools of thought exist.
New rings every X cassette replacements, and new cassette every Y chain replacements. This required consistent records kept up to date correctly. Nowdays this is much easier with software like Strava, provided you set up the parts on your bike.
This is the more-expensive method in the short term, because you are replacing parts while they still have some wear left in them. This leads to better performance overall, so would be the racer's preference.
Ride it and change parts when the function stops being reliable. This is more common on bikes used as transport rather than exercise or racing.
Also if I have a used bike with an unknown mileage on its parts, then I might take the extreme course of riding till it fails completely, and then replace all the transmission parts (cassette/chain/jockey wheels, maybe chainring if needed, along with inner/outer cables, brake pads, bartape/grips)
This tends to cost larger amounts but less-often.
Overall, either plan still costs less than driving a car the same distance.
Personally, I rarely change chainrings because they wear much slower having so many more teeth engaged. I tend to change cassette along with a new chain.
No! No! No! A thousand times no!
A worn chainring engages to a new chain (and a worn chain too).
A well-worn chain does not engage to a new chainring. So in fact the converse is true: you don't want to replace chainrings when replacing the chain, but you do want to replace the chain when replacing a chainring.
I have a crankset with visible shark toothing in the chainrings even though the chain wore to just over 0.5%, never reaching anywhere near 1.0%. The chainrings work just fine with a new chain.
There are two reasons you might want to replace chainrings:
Wear. Good big (50 tooth) chainrings made from highest quality 7075T6 or 7075T651 aluminum can last perhaps nearly as many kilometers as a typical car engine. There used to be a picture by Jobst Brand of a chainring that had seen just this mileage. The teeth were very worn but the chainring still barely worked. The chainring was rotated 72 degrees several times in its lifetime (it was a 5-bolt chainring and not a 4-bolt that would support 90 degree rotation) to even out wear. Unfortunately, the picture is no longer online. However, this demonstrates that good big chainrings do indeed last a long amount of time. A single 1.0% chainwear incident is not a reason to replace chainrings.
Chainsuck. In some cases, small chainring used in bicycles with long cage derailleurs (mountain bikes and hybrid bikes) can experience chainsuck where the chain does not disengage from the chainring as it moves towards the rear derailleur. The only solution to this phenomenon is replacing the chainrings. On road bikes where 99% of the time the big ring (50-53 teeth) is used and the derailleur cage length is short, this never happens.
However, before you claim that your chain is 1.0% worn consider the quality of the tool you use for measurement. Today, the only reliable chainwear tools are Shimano TL-CN40, TL-CN41 and TL-CN42 which are simple go/no-go tools telling whether the chain is worn more or less than the limit (probably 0.5%). See here for details. I know of no tool that would tell 1.0% wear that would be reliable, apart from an inch ruler longer than 12 inches (a foot).
So if you are not measuring your chain with an inch ruler but rather using a tool that tells 1.0% wear, your measurement is probably crap. Discard the tool and buy a Shimano TL-CN42 instead.