I'm not from Texas, so maybe there are no local issues there in which party policy makes a difference. There are places where it does, and places where certain policies simply cannot be credibly espoused by both parties. Even in small towns things like education funding can follow party lines. You yourself cite Mamdani's winning policy. According to your logic, a Republican candidate should have said the same things. But it's unlikely such a creature would be nominated, funded, or supported by the party.
Sometimes it might depend on how slavishly the local Republicans follow the national party line and policy. In some districts, a reasonable policy on local matters might be largely shared by both parties. In the small towns where I've spent most of my life, including a good bit of local political life, there's a certain flexibility in the minor positions, where fitness for a job outweighs nominal party membership. Town Clerks, effective committee chairpersons, and some such positions, are sometimes double-endorsed, but not likely Selectmen.
But there is still a symbolic and philosophical difference. And these days, national policy is bleeding into the local world in ways that become pretty hard to ignore. If you live in a Vermont community, for example, even one that's usually red, and masked, gun-wielding ICE agents are smashing the windows of commuters' cars, bursting into houses, carting long standing community members off in a blaze of violence and obscenity, and other policies have defunded food and fuel assistance, and harmed working farmers, perhaps party membership will seem relevant to some. Again, I imagine that varies with region. Here in little old Vermont, we can still have a Republican governor who openly disagrees with much national policy, and has enough popularity and clout that he has managed to hold off the open desire of the States R. committee to oust him. That's partly perhaps because they know that no candidate they prefer could ever win here. But I don't think that's true everywhere.
Some things take time, too. It may take time to win, and prevalence can be cumulative. It seems odd and frightening for an American even to be thinking of words like acquiescence and collaboration, but I believe it's meaningful these days, even if you have no present power, to assert a dissenting opinion.