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This should help:

2. Select the device you’d like to sign out of.

3. Select Sign out.

You can also remove YouTube on TV access for a Google Account by opening https://myaccount.google.com/permissions > select YouTube on TV > Remove Access."

I'm guessing this smart TV is using some kind of weird nonstandard auth scheme that has fallen through the cracks.

Maybe it's storing some kind of non-expiring login token that isn't invalidated using the regular process.

I've done both of these things - to no avail! This is really weird

Did you already mention the brand/model/year of the smart TV or the OS/Platform/version etc

Maybe that might narrow down some search results to p3culiar issues seen only on that firmware or app version? Just maybe