Yeah... some early implementations just allowed for any authority to be set in the header and trusted it... that's of course wrong from the start... if you only allow for trusted or "known" authorities a lot of the contextual concerns become non-concerns.
Beyond this, you can make shorter lived JWTs just fine in the browser and have the agents self-update. If you use Azure Entra or a number of other providers it works this way in practice... you keep your JWTs relatively short lived (5-15m) and can even check for jti revokation.
JWTs are incredibly useful for separating/reusing an access authority from your applications/api systems. You shift the attack surface and do it in a way that can be trusted. We use PPK for lots of things, including SSH all over the world. No, I wouldn't use shared secrets and I wouldn't use long lived tokens... but short lived, ppk signed tokens from verified/known sources are generally fine.
For that matter, it's often API keys that are really problematic. Just had to implement them... for me, the API key presents as a Bearer token as well, but there's a short "sak." prefix then an identity part (base64url uuid bytes) followed by a secret as base64url bytes... in the database is the uuid and a passphrase level salt+hash from the secret.. so the api key generated should be treated as a secret and is one-way to the database, so a db breach doesn't breach auth.
Even then, an API key leak is far mroe likely than a problem with a well implemented JWT solution.