Thank you for all the details, it helped a lot!
I think I found the conflict. I am not entirely sure what the exact line of cause is, but I narrowed it down to this option on the Index Table options panel: https://i.imgur.com/3pBwZQY.png
I assume, that maybe this triggers the indexing too many times, and some programmatical hooks might be left deregistered afterwards, or the function terminates at some point, and then on the next pageview it finally executes.
Anyways, I have left that setting deactivated, can you please confirm, if that fixed it?