>The problem is when cloudflare doesn't let you through.
Don't use an unusual browser configuration then, like spoofing user-agents or whatever? If you're doing it for "privacy" reasons, it's likely counterproductive. The fact that cloudflare can detect it means that the spoofing isn't doing a very good job, and therefore you're making yourself more fingerprintable.
There's a whole lot of things that can count as "unusual" that aren't spoofing, and telling people not to be super vague "unusual" is a terrible solution.
Ad block, other blocking, third party cookie restrictions, all the stuff firefox changes when you toggle resistFingerprinting. From your other comment "users with no google cookies" and "connecting from VPN".
Punishing people for not having Google cookies is probably the most obnoxious one.
Don't use an unusual browser configuration then, like spoofing user-agents or whatever? If you're doing it for "privacy" reasons, it's likely counterproductive. The fact that cloudflare can detect it means that the spoofing isn't doing a very good job, and therefore you're making yourself more fingerprintable.