Hi Australia,
Replacing the battery was the right first step; you've eliminated that as the source of the problem by doing so.
Ordinarily I might suspect the charging port, but two things are saying no, that's not it. The fact that it shut down with 40% charge left on the battery tells me it's not the amount of charge left in that battery that was the problem, but it could have been the Battery Management System, the little board mounted on the battery that holds the temperature sensor and charge indicator. However, replacing the battery should have eliminated that as a possibility.
Also, replacement batteries are usually sent out with some charge in them, so plugging in the new battery should have given it enough power to at least turn on. So no, it's not the battery, as you've already concluded, and probably not the charging port either as we just discussed.
You can double-check that it's not the charging port by setting the phone down on a wireless charger. The wireless charging uses a different circuit and different motherboard connector, so it should still work even if there's a problem with the USB port. I don't really expect that you're going to find any difference, but might be worth a few bucks to find out.
So, as you've surmised, what we'd be left with is a logic board problem. It would probably be more straight-forward to get the existing motherboard fixed rather than try to swap memory chips with another motherboard. Newer phones are encrypting the memory chips and tying them to the CPU so that if you move them to another board you won't be able to read the data. I don't know for sure if they're doing that with your phone, but it is a growing trend and a distinct possibility.
I'd say your best chance is to try and find a repair shop that's experienced with board-level repairs on Pixels. That's going to depend a lot on where you're located.
Good luck; let us know what happens.