We were experiencing the same problem. After more than an hour of figuring out why the select and date input didn't work, I found out that it did work correctly in Safari. This bug is still present in iOS 16 beta 2.
Just as Cabanossy mentioned: it indeed seems that the select fields / input fields do load something because there are several 'dark' spots on the screen, even after switching between pages. The date picker also randomly appears at a random place on the page.
We use Vue in a SPA, maybe this is useful debug information for Apple.
Hopefully there will be a solution soon, the customer was really happy that they could add their web application as an app.