How does this work in your head? When the new homes are being built, where do the people who were living on the land they're being built on go?
A 500k house never gets torn down to become 3 400k condos. It becomes 3 600-800+k condos.
We should build more, because we need it. But "build more" can't be the only tool to lower prices, because simple observation shows us that it doesn't work.
...which explains a lot about why residents dislike this
Quite the contrary. A pretty simple reductio proof demonstrates that more housing lowers prices. All you need is to imagine a market in which there is one more apartment available than there are people to rent it.
Notice how these responses do absolutely nothing to address the issue. They exist only to express dissatisfaction with realistic options.
Again, you can either build to meet demand or expect higher prices. Those are the options.
Cities have been able to grow for centuries. The only thing stopping them from growing are zoning and restrictions prohibiting additional homes being built.
Even if a city government has liberal attitudes toward development, it still takes years to create new supply. In the mean time the process is displacing people.