By the pigeonhole principle not everyone who wants to live in a growing city can own a house. So long as houses are allocated economically (as opposed to submitting an application to join the queue at the People's Bureau of Housing or whatever) this will reveal itself as some people being unable to afford houses. So we can conclude that the affordability of single family homes in isolation isn't the problem we should focus on.
> So yes, things become cheaper if everyone wants to live in an apartment.
People make tradeoffs all the time. Lots of people would love to own a nice house in a nice neighborhood within walking distance to a forest and to downtown for not very much money. But too many people want to live there for them to all own houses. So some will choose to live in an apartment instead, and others will choose to move farther away to be able to own a house. Preventing upzoning will just mean that fewer people get to make the tradeoff to live in that neighborhood, and only the richest of them will be able to pay through the nose to do so.
> But for modest income folks in small old houses, up zoning means moving out of the community.
Well, nobody's forcing them to. After their house has been upzoned they can live in it for as long as they want, then sell it to a developer for millions of dollars when they want to move. The developer will use the land to build apartments, and sell them to people who don't have a million dollars to buy a house. Everybody wins.