You had me curious. I know k-9 has specific push settings, so surly the wouldn't remove push. Well, it seems Thunderbird relocated or simplified the settings, but it still exists.

First, you have to enable the alarm permission for Thunderbird/K-9. You should have a TB notification telling you the push service is running (which you can hide). Also make sure "background usage" is enabled and no battery optimization is on

After a bit of testing, I have got some odd results that seem to point to deeper issue with android, probably some odd permission or setting screwing everything up.

My test was simple, I had my account set up on both K-9 and TB, plus the gmail app as a control. I sent an email to my self from another device and waited.

Basically all 3 failed for me. Nothing happened until I unlocked the phone (yes lock screen notification are enabled and other stuff works fine when locked)

Even then, results were mixed. The first time k-9 came through, while gmail and TB did nothing. Another test and gmail showed up first, another and all 3 showed up etc.

Quite strange, but yes TB does have push notifications. It seems the way android does permissions these days, they have to do some whaky shit to run the service. I wonder/suspect its bypassing google play service and maintaining a server connection to accomplish this (based on the need to set an invisible "alarm" and setting "Refresh IDLE connection" just below the "Max folders to check with push" setting).

It also seems how I have things set on my end, my push notifications are fudged up across the board....