Well, no input from the group, so I ended up taking the Ody to a mechanic. He said they had to clean out the EGR passages, which were nearly blocked off. (I still have no idea where these passages are on a 3rd gen Odyssey... but there you have it.) Once the passages were cleaned, they put on a new EGR. It didn't fix the issues. (which makes sense since I'd already tried two new EGR's.) They realized that the signal between the EGR and the ECU might be a faulty ECU, and replaced the ECU with a used one off e-bay. That fixed the issue, the check engine light cleared, and I was able to pass DEQ and get the car registered. All in was $100 for the used ECU and 2 hours labor. Not too bad.

If anyone else needs this same repair take note that the ECU's are VERY specific to the model of the car. You'll want to pull your ECU, check the exact version on the sticker, and replace with the same kind. They also had to do an update, which I think put the ignition/key information from my old ECU onto the new one. Good luck if you have to do this yourself. I'd probably swap the ECU and have the shop do the last steps.

 

I appreciate the followup. I have a 2016 with the same code (P2413) and no improvement with a new EGR. I'm not entirely clear on how to clean out the passage(s) but it sort of looks like I have to pull the intake manifold and then hack at it with a coat hangar. Or just take it to a mechanic instead of standing in the weather for hours trying to do it myself.