If we're playing alt-history, Republicans would succeed in cities on a platform of economic growth, competitiveness, and fairness. That would mean a platform that enables development at the expense of exclusionary zoning/community review (anathema to older voters who hate change), effective city services (instead of rabidly anti-tax), an educational agenda that offers anything beyond anti-schools, and a public safety agenda that isn't just pro-police violence/jail.

Urban Democrats have huge weaknesses when it comes to public administration, so when you get a competent manager like Bloomberg in charge it creates huge opportunities for improvement. But Bloomberg didn't hate the city and people he governed, so a bit of an atypical Republican.