It's not Firefox that's the problem; reCAPTCHA works just fine on Firefox. It's all those anti-tracking measures you installed and enabled -- they work by making your browser indistinguishable from a low-quality bot, kicking the website into self-defense mode. The slow fade is a rate-limiting measure. It's annoying to you, but it's more annoying to people trying to automate login attempts.
The site is attempting to protect your account by preventing automated attacks against it. Meanwhile your browser is doing it's best to look like a shell script, refusing to send any sort of behavioral feedback or distinguishing characteristics that might give away the fact that you're a human.
So the question is: is it really worth alienating those quirky, paranoid users who take extraordinary anti-tracking measures, just to protect your normal users from automated attacks?
Yes.
Of course it is.
Incidentally, you still need rate-limiting if you use Google's CAPTCHA. If you don't rate-limit CAPTCHA endpoint, an attacker can DDoS you (especially if your server-side captcha component uses low-performance single-threaded HTTP client). Furthermore, an attacker within the same AS as their target can purposefully screw over their account by performing attacks on Google's services until the reputation of the network hits rock bottom.
Conveniently, normal users with typical browser configurations get nothing but the animated checkbox. For nearly everyone, the whole experience is simple and easy. The only people who get inconvenienced are the low-grade privacy enthusiasts who think that preventing tracking is the path to Internet safety. Ironically, "tracking" is literally the mechanism by which legitimate users can be distinguished from attackers, so down that road lies a sort of self-inflicted hell for which the only sensible solution is to stop hitting yourself.