I just set up my servers' SSH to use U2F keys for authentication and the secret never leaves the security key, protected behind the security key's HSM.
That's kinda the point of these security keys: computers aren't devices to be trusted. If they were, we wouldn't need physical security keys with the secrets hidden behind a HSM.
From TFA:
> Local threats like JavaScript injection attacks could exfiltrate the value of inputKeyMaterial from Step 2.1 and store it away for later use.
I really enjoyed TFA but... These kind of "local threats" are precisely what these U2F security keys are all about: you can use them on a totally compromised computer and yet the attacker cannot extract the secret from the security key.
I've seen people use security keys with GPG and I'm nearly sure the private part of the key is never leaving the security key.
I may be wrong though...