From what you have said, I do think where there could be a lesson in what happened to Davis comes down to authenticity; he came into the race as the kind of Republican that shouldn't have too much of an issue winning a place like Jacksonville and, because of the demands of a fractious primary process with a MAGA candidate, ended up completely shedding his brand while turning himself into something that he wasn't.
Just from my experience who (at least for the time being) works in and resides nearby to Fort Worth (soon to be the largest city in America with a Republican mayor), the fact that remains the case is because the Mayor is very moderate (to the point of being openly critical of state government at times) and has gone to great pains to carry on the more moderate image and appeal of her predecessor, who she used to work for. If she did a heel turn and did what you have described from Davis, she'd be at serious risk of a credible Democratic challenger and absolutely could lose. The same is true in Arlington, where I live.... the past two Republican mayors have even participated in events with the Biden White House surrounding more bipartisan issues, such as infrastructure investment in part because in the city they represent, it's good politics for them to present a more moderate or bipartisan image.
In any event, it's a tough balancing act for Urban Rs these days, probably not unlike the issues that Rural D pols can have.... their goals and the goals of the national party are often at odds.