The problem is, the button to start a video conference is hidden (by default) if you are using a mobile device to view the site. I have the latest Samsung Galaxy, browsers that are compatible so, I am not sure what you mean about "unsupported mobile browsers".
As of right now, if a Desktop user created a group video chat, it sends that push message to those people with a red join button. If anyone who gets that alert and is currently viewing your site with a mobile device and clicks join, they are prompted to download the Jitsi App and then it connects them to the room.
You mentioned adding a way to have your users download the Jitsi App some other way. That still doesn't fix the issue. Reason is, the button still will not be available in your site. If you open the Jitis App directly, well, you just created a room where you are the only person there, no push notifications, and have no way of getting others to join without posting the link somewhere on your site.
What I am thinking is, having the button with an option to enable/disable for Mobile browsers so not everyone has to use it. If someone viewing your site with a mobile device creates a group chat, then wants to start a video conference, a popup would appear to download or continue to the app (like it does now). BUT at least this way, you get the hashed link, the push notification would go out, and others at the site will have the option to join.
As I mentioned above, not everyone has developed/published UNA Mobile Apps for their sites.