Hum, I'm actually developing und testing this Firefox first. – Maybe a case of bit gremlins in the wire? (Try a force reload: CTRL/Command+SHIFT+R.)
Also mind that there are certain differences, when using "Edit mode" and "Games mode". (E.g., in edit mode, the shift keys are sticky, in games mode not.) Moreover, in games mode, the CAPS LOCK key of your physical keyboard acts as a toggle for virtual joystick and numeric key block mappings.
On Firefox I get the "READY" but am then unable to use the keyboard or the drop-downs. When I click on a drop-down and wait for half a minute it might finally show it, but it's unusable. Firefox also regularly comes up with a warning that the page is slowing down Firefox, and the browser indeed seems to be hogging a single core.
This is in Firefox 126.0 on Fedora 39.
[update: I tried with safe-mode to disable any extensions but this doesn't make a difference]
(This is hard to overcome: the emulator has to render in a 60Hz duty cycle and we're emulating a CPU at 1 Mhz. If sound is enabled, we also have to sample sound at 1 MHz, and resample this to 48KHz digital audio.)
There seems to be some serious misconfiguration at play, though. For me, this runs on the oldest modern machine, I have around, a 2008 MacPro with the original ATI 256MB graphics card, under Firefox 78.15.0esr quite perfectly. This is probably puny in comparison to your setup.
I guess, inadequate hardware acceleration may cause similar symptoms. I've seen this in the past (and in different context) with Chrome on older machines, where the frame rate drops to something like 2 or 4 fps, while it keeps up perfectly with hardware acceleration disabled. May be worth a try, maybe FF and your GPU driver do not play well together. Compare: https://support.mozilla.org/en-US/kb/performance-settings