If you set protection settings all the way up in Firefox, Cloudflare will get stuck, yes. I do not consider this a shortcoming of Firefox though, my browser is doing the right thing.

Previously, I kept a Chromium instance around, but now I use a second Firefox profile that still has tracking protection enabled, 3rd party cookies restricted etc., but set to a level where e. g. Cloudflare still works. I trust Firefox to better act out my interests. Using -P, the second profile can be started without launching Firefox's profile manager.

With browsers beginning to actively fight fingerprinting techniques, I expect more sites (legit or not) that rely on these techniques to break. That is fine by me.