Free software does not mandate service operators to interoperate between their own infrastructure and custom or modified software you possess. You are not somehow guaranteed access to someone else’s system just because your local code has a FOSS license.
The latter is what I mean. Web Integrity or SafetyNet are a closer counterpart to this situation than your curl example. A user agent is just a hint; attestation is an effective enforcement mechanism.
No Free Software license on my client can obligate your server to cooperate. But if attestation becomes the norm across many different services, the entire framework of user choice Free Software stands on collapses.
How would Ungoogled Chromium survive if every second website rejected non-Google Chrome? We've already seen this play out with Android: more people daily-drive GrapheneOS than Lineage because it passes Google's lower-level integrity checks.
And when a "strongly advised" government app requires the strictest attestation level, will you have to move countries just to exercise that freedom of choice you supposedly have?
In the same way that commercial entities aren't guaranteed a business model just because they want one, free software isn't guaranteed connectivity to other people's systems just because otherwise it's less useful.