On my Samsung S24 (Android 16, oneUI 8.5) and A53 I get no background synchronizing of my POP3 account. I guess I have set all options like battery usage, mobile data usage and what ever I found in the support forum. The behavior is the same for K9 and Thunderbird.
Now I found a workaround: I created an additional imap account that I configured in push mode. As long as I run it in push mode, the POP3 accounts a synchronized every 15 minutes. Seems that without imap push mode no one wakes up K9 in background mode.
I guess most K9 users have at least one imap account and do not recognize the problem but I normally do not use imap and especially no push mode since it encreases power and data consumption.
Hopefully some one can solve this problem.