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Edit: I figured it out. It was caused by the User-Agent Switcher and Manager addon, I use it to switch user agents on one site. I switched it to whitelist mode in the settings and now it works.
Like when I try to log into chat.openai.com, I click the checkbox that I'm human and it just keeps reloading that checkbox without progressing the page. But when I open the tab in a Firefox container (which you can install with the Firefox Multi-Account container addon), it works.
I want it to work without a container open.
Guesses: VPN, AdBlocker, Browser profile spoofer (chameleon extension f.e.), fingerprinting protection (Firefox's or from extension). If you have something from this list, try to disable it and go to site.
VPN alone shouldn't be affecting Cloudflare turnstiles, but any extension meddling with the user agent and headers will. You're on the right track.
Ding ding ding! User agent switcher addon. See my post edit.
This! 😄