A longtime nickname among Cleveland fans has been The Tribe, which i think would be better but still not a clean break and still problematic as regards team logos.
Cleveland does have a set of iconic statues that are officially called “Guardians” (specifically, the Guardians of Transportation), and everyone recognizes them, but nobody ever calls them that. If Clevelanders ever have need to refer to them, they’re “those things on the Lorain-Carnegie Bridge” (the bridge itself is also never called by its official name, the “Hope Memorial Bridge”).
That’s part of what makes it good. Sports teams are supposed to be named after something the other guys fear. That’s why so many are predatory animals.
Dodgers would just avoid 'em.
Giants are too big to be bothered.
Rockies are too hard to be bitten.
Blue Jays, Orioles and Cardinals eat spiders.
Astros and Angels are above it all.
Nothing bothers a Ranger.