No, we can't. Once could lower the rate of increase of housing prices (and, more realistically, land values), but "gentrification" isn't really just about pricing, but a change in preferences: Areas close to urban centers had low prices because people wanted to live in deep suburbs, far away from city crime and bad schools. Now they are desirable, and thus people far more affluent are going to bit outbidding the communities that live there anyway. So the gentrification is inevitable, and IMO any attempts to stop it are bad: Nobody has right to a character of the neighborhood, whether we are talking a Mexican American community or people that want racial covenants in their subdivision. The entire idea is just straight out illiberal in the broadest of senses.

It's true that the gentrification process speeds up when there is no building, as those differences in preferences mean existing tenants are competing with even richer people, and would have to move even further out, but the change in preferences still exists, and thus the gentrification is straight out unavoidable.

Besides, in almost every place where we have serious housing problems, the small changes proposed here are insufficient. When the land is expensive, the normal behavior everywhere is to redevelop plots to the maximum economically viable density. Turning 1 house into 4, which would sell for 500k each, doesn't make sense when the possibility of an apartment building is there. Single stairway apartments turn those 4 cottage homes into 16 3-4 bedroom units. If one is simplifying permitting anyway, why not simplify it for that density, and get far more out of the same land. Those 4 new cottages, now new, would not get redeveloped again for another 30, 40 years, so they would become difficult to afford really quickly. That's why you don't see many places actually attempting this low key densification, as it's way too much work for what you get.

Rights != Desires.

Desires are much more relevant in day-to-day politics. Rights are less frequently up for question.