Greetings,
tab-bar-tab-post-open-functions need to be called during tab-bar initialization. Currently, a user's custom post-open functions are invoked only for tabs created beyond the initial tab. This is a bug, in my view.
I've created the patch vs. this morning's master and should also benefit Emacs 30. I've been running a patched copy against my daily-use 29.4 for a bit and it works as expected.
-Stephane