Hi everyone, greetings.
For this issue, I just got the solution. Maybe it works on you as well, mates.
So I am using Samsung S23 Ultra with Android 14 and face the same problem.
So my solution is: instead of using Chrome to install the App, I tried Mozilla Firefox for Android and did the same way as Chrome to install;
which is clicking ‘Add App to Home Screen’, then click ‘Add’.
When I access the installed App (from Firefox), the pop-up to choose Camera or Media is appearing! I don’t know if there is a different method of installing from Chrome and from Firefox, but it works for me.
And please don’t forget, before installing your App using Firefox, be sure to uninstall it first (the Chrome-based installation) to be replaced by Firefox-based installation. Hope works for you all as well.