Checking a box is fine when it lets you through.
The problem is when cloudflare doesn't let you through.
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.
Examples?
Punishing people for not having Google cookies is probably the most obnoxious one.