I was able to reproduce it and pin down the problem.
Everything above is still true. You can start with all Cache and User Data cleared from Vanadium when trying to reproduce. I altered No settings but gave permission for notifications (has No effect probanly)
Go to a Website (Not in private) that you can Install as an "App"
Install
Open the "App" from homescreen
From now on swipping to Close does Not clear private browsing Data for me
I tried two Websites both have the same effect
One is https://www.kuketz-forum.de/
Maybe someone can try to reproduce
Pixel 6 btw