Frankly speaking, it's undoubtedly the worst TPOW role at the moment. It's in an awkward position where it doesn't feel good enough to be a Town Power role, however at the same time it's too prominent to be a Town Support role.
This brings me to the rework itself. I believe Mayor could function in a similar way to the Judge. The reveal function should not be name-bound.
To put it simply: revealing as a Mayor makes the town aware of your existence the same way calling court as a Judge does. The Mayor would be able to alternate between chatting publicly under their own name and chatting anonymously using their role moniker. They would also be able to alternate between a personal single vote and special triple vote. This would reward quick-witted and smart players while making no difference to subpar players.
(Edit: ignore alternating between votes)