We provide an endpoint to check validity with the server, but haven't used it too often. Anything "reasonably sensitive" (or more) doesn't depend on anything like this client-side security.

But, if you're just hiding an additional Delete button on a page based on claims, this comes in handy.

(Edit: in one case, we've used asymmetric keys, i.e. public key so everyone can check integrity. This was a very different use-case to most web apps, though. Overall I'd say if you're carefully checking integrity of something in client-side JS to do something, I think that's probably the wrong approach)

That's exactly what is useful for. Of course access to a resource is determined server-side; JWT simply allows you to adjust the UI to the permissions the user has without any additional calls. If the user changes the JWT he has client-side, he will just get a broken delete button (the server will reject a JWT that has been tampered with).