Dual Citizen Transiting Through the Schengen to the US—Which Passport Should I Use?

The easiest would be to use your US passport all the way through.

The only downside is that you can't use the Schengen lines (or automated control) at immigration in Oslo and Keflavik, so you may have to wait a little longer.

Since Icelandair is a low-cost carrier, you will likely check-in online for your flights. Check in with the US passport and show that to any gate agent. You can (see below) present the US passport, but present the Polish passport at the Schengen border controls would make for an assuredly smooth experience.

Or am I legally required to enter the Schengen and/or EU on my EU passport?

No. Until the ETIAS travel authorisation is implemented sometimes next year, you can enter/exit with any passport you hold. Poland and the US allow dual-citizenship, you are fine

If you show a US Passport at the exit controls where the stamps would indicate an overstay (you were in Schengen for more than 90 days in a 180 days period without an appropriate visa/residence permit), just show your Polish passport (or even ID Card) along with it and you will be cleared.
The same applies if your US passport doesn't bear an entry/exit stamp because you last entered/exited on the Polish one.

Will the border authorities be confused as to why some John Doe from the US never crossed the border in OSL/KEF despite arriving on a flight but why some random John Doe from Poland did even though they weren't on any flight manifest?

No. until the Entry/Exit system, which is a Schengen wide entry record database is deployed (again sometimes next year, along ETIAS), your entries as a third-country citizen are materialised by your stamps, but see above for that.

The carrier may want to check at the gate your permission of stay (Ryanair is infamous for doing this), in this case, show the Polish passport.

Which passport do I show the airline at check-in in LYR?

Show the passport you will use at your final destination, the US passport. Once ETIAS is in place, you'll need to show both passports for a trip like this to avoid paying for ETIAS authorization to enter the Schengen area with your US passport.

The main reason I'm posting this answer, though, is to respond to some of your comments:

it's a short stay, so I guess it doesn't really matter

You'll probably get through more quickly with the EU passport, in most cases much more quickly. There's also a more serious consideration, although for most people it is of no practical consequence: your right to enter the Schengen zone is greater as an EU citizen than it is as a US citizen.

For example, if you do not show your EU passport, the border officer can question you about the purpose of your trip, might decide that you are not credible, and might refuse to let you enter altogether. So in fact, you are arguably doing a discourtesy to the officer by requiring the additional work of evaluating your application for admission as a third-country national when in actuality you are an EU citizen. (In fact, if there are automatic passport control kiosks you won't have to take any of the officer's time if you show the EU passport.)

I wasn't sure if switching passports between flights made a difference

It doesn't. You can always update your passport information. And using one passport to clear transit border controls when you've checked in for the flight with another passport has never been a problem for me. If it ever is a problem because of an inability to match the flight manifest, just show the other passport as well.

it seems easiest to just use the US passport the whole way through to save my brain the mental gymnastics.

For me the bigger problem is the fumbling. Mentally, I find it simple to remember that whenever I'm showing a passport I should choose the one that carries with it the greater rights.

Seems that the requirement for many EU citizens to use their EU passport for entry/exit is really only extended to situations where you're entering/exiting your country of citizenship (i.e., a Polish citizen entering/exiting Poland), not for other EU/Schengen countries.

Yes, because these requirements are imposed by national law rather than by union law.

My US passport has plenty of Schengen stamps from my pre-dual citizen days, so it's not like it would be suspicious for me to use it for entry anyway.

On the other hand, the many stamps might just arouse suspicion that you have been living in the Schengen area without authorization. Certainly, if you show the US passport and they start analyzing the stamps, you would at that point want to show the EU passport to save everyone some time and effort, wouldn't you? Why not just start off with the EU passport?

When talking to someone, you can show both and stop worrying. You may be asked to enter your details in advance and will indeed need to mention your US passport to the airline because you don't have an ESTA or visa but other than that you cannot do anything wrong.

On the Schengen / Norwegian side, it really doesn't matter. I would use the Polish passport at the border itself but there is no requirement to do that and no way you can get in trouble. Nothing will be matched automatically and even if and when it is (the future Entry/Exit System or if a guard asks about entry stamps in your US passport), you can always show the Polish passport and end the discussion. There is also absolutely no problem with showing the airline the US passport and switching to the Polish passport at the border (if only to go through the “EU passports” lane).