Camera and mic access in Doxy (Chrome and Firefox)

I'm using MacOS Ventura; Firefox 199.0; Chrome 109.0.5414.87 on a 2018 MacBook Pro (Intel)

I am posting this question here and with Google, and Mozilla because I’m not sure where the problem is. I don't think it is Doxy's because Doxy assumes the connected OS and browser are controlling the camera and mic.  The common factor is MacOS 13.1

For those who don’t know, doxy.me is a "Zoom-like" secure telemedicine platform.  Doxy advises users on Macs not to use Safari, but to use Firefox or Chrome.  I can attest that Safari and Doxy do not play well together.

Logging into a regularly scheduled meeting today, I found that Doxy had no access to my camera or mic. In the past, I have not had this situation.  

I did a number of things on my Mac (restarting the app, restarting the machine, even resetting my SMC at restart) and none of them seemed to work.  

I had already checked that the OS gave Firefox access to the camera and mic, and Firefox's settings gave camera and mic access to doxy.me.  After many attempts, I was able to get into the meeting albeit 35 minutes late.  

As soon as it ended I retried getting in and found that access to the camera and mic were again disabled.  I have found a workaround that involves toggling the camera and mic permissions in the active session window. Once that is done, the camera and mic work. But the problem recurs if you quit and restart the app. Looks like a Firefox problem.

While I was trying to get into my meeting, I tried Chrome and got the same camera with a line through it. Later, I found similar controls in Chrome that allow for the same workaround as in Firefox. The problem recurs after quitting and restarting, but only about 15% of the time.

This makes it seem that Ventura (or the Mac hardware?) is not always honoring the permissions set in two different applications. Hardware is stupid and the OS is supposed to be in charge.

Anyone else seen or having this kind of problem?

P.S. I have not experienced anything similar to this in either Zoom or Skype.