Update: this has been meanwhile fixed by Cloudflare, see [1] and [2]; I can also confirm that turnstile now works here with SeaMonkey.
[1] https://forum.palemoon.org/viewtopic.php?p=249856#p249856
[2] https://community.cloudflare.com/t/browser-integrity-check-b...
Sadly comments like [3] are realistic observations here; as I said, it's not the first time this happens. At least the support conversation at [2] isn't full of "your browser is outdated" like [4] from 2022 (when Cloudflare's servers were requiring Origin: on all requests, which I think is also what the hackernews thread at [5] was about).
[3] https://forum.palemoon.org/viewtopic.php?p=249864#p249864
[4] https://community.cloudflare.com/t/browser-integrity-check-b...