I had a massive issue with dozens of thousands of bot accounts over past months, so relying on reCaptcha seems to be the only way fighting this. Unfortunately it looks like this extension is indeed causing major side effects, making parts of the forum functionless. In the past, at least for me, it caused issues when registering via 3rd party logins.
Now, I can reproduce the issue as @agpx stated:
Nothing useful in network tab: (not very surprising)
There is no particular error in flarum*.log file.
EDIT: A (temporary) solution/alternative might be using Turnstile instead. This works: