dual us citizen with about to expire us passport, checking-in to departing flight leaving us with foreign passport

1. You always check in with the passport that you will use to enter the destination country
2. Always enter and leave a country with the same passport
3. If you switch passports during travel, let the airline know, especially if the country has exit controls. Example "I'm leaving Germany on passport A and will enter Korea on passport B". The US doesn't have exit controls, so that doesn't apply to you.
4. Security and TSA only require a valid photo ID. They don't care which one, even your driver's license will be fine.

In your case, you will have no problems leaving, but you may have problems coming back. You do need a valid US passport to return to the United States and you will have trouble trying to board a US bound flight with an expired passport. There currently is a potential Covid exemption at the moment but this may go away anytime without advance notice and it's not clear that all airlines apply it.

I almost always check in for flights leaving the US with my non-US passport. I show that passport to everyone, including the TSA (who are only identifying you, not investigating your immigration status). It's never been a problem. I have the sense that some US airlines may pay attention to the immigration status of departing foreigners, but I usually use European airlines to leave the US, and, so far, they haven't cared. If anyone ever did, I would show my US passport.

The only person I would expect to require that I show a US passport would be a CBP officer doing departure spot checks. These do happen, but I've never seen them.

US government sites, as you note, typically misrepresent the statutory requirement. That's not your problem, though. There is no requirement for you to check in with your US passport when you leave.

(This loose interpretation of the law might be a problem if there were a penalty or some other effort to enforce it, but since there is not, you can just ignore it: there's no chance whatsoever that you would have to go to court to defend your interpretation.)

Is 'checking-in' to a flight with a foreign passport leaving the US even though you are also 'bearing' a valid US passport still considered illegal?

Why would it be? As you note, you are "bearing" a US passport. That the administrative systems make it difficult for you to provide evidence of that fact through your passenger record also shouldn't be your problem.

I do not know whether there is any way for the airline to list multiple passports on your passenger records. They potentially have to send passport data to two governments, so they may be able to put different passports in each message. This may become more of an issue after ETIAS goes live; the UK has also announced a similar system. Today, a dual US/EU citizen can check in for a flight to the Schengen area with the US passport and then show the EU passport on arrival. With ETIAS, it would be necessary to pay for the electronic travel authorization before doing that, which shouldn't be necessary.

In a comment, you ask

After checking in with my foreign passport, will it cause me trouble if I show my American passport or both to TSA?

TSA doesn't really care what kind of ID you show, though I was once told I could not show my driver's license because I was flying internationally. I showed my foreign passport. In any event, TSA does not check that you're using the same document that is recorded in your passenger record.

Also how do you know who is TSA and CBP?

They have uniforms with badges and patches that identify them. But there's no reason to avoid showing your foreign passport to anyone. If that person turns out to be a CBP officer, he or she will probably ask about your immigration status, at which point you can show your US passport.

Does it make your life more complicated or less complicated if you just show everyone both passports?

I haven't tried this, but it's probably more complicated to show two passports when there's no need to. It shouldn't be very much more complicated, though, so why don't you try it and come back and post an answer to describe your experience?

Yes, you can most definitely do that (have done that too many times to count). The information you provide to the airline is mostly for the destination country, not the one you are leaving, and since the US do not do any exit checks, there's no-one else to show your US passport.

At some point in time check-in personnel would ask questions when you left the US with a non-US passport after having entered with a US passport (mostly back in the days of paper I-94 forms, when they failed to find the form in your passport), but indeed showing your US passport (or even just stating you had one, I don't remember) closed the issue.

It makes it a bit more fun for CBP/DHS to match your records, but that's their problem.