I figured out why this is happening.
Plume uses nostr-login, which inserts itself between the page and your signer so it can pass messages along, instead of talking to the signer directly. Sidecar doesn’t support that, so the login button appears to do nothing.
The nostr-login library comes from nostr.band, a service that has shut down and is no longer maintained. I’d suggest replacing it with a direct connection to the signer extension through window.nostr, which is how most modern NIP-07 clients work.