Because it doesn’t drive down prices.
There are also less visible efficiencies in having people living close to their work, like roads that aren't crammed with cars because people don't have to spend three hours behind the wheel every day getting to and from work, and the additional productivity that comes with all that extra time.
What I'm not seeing here is a way for increased housing supply to make the prices go up, as opposed to down, but not as much. Network effects, maybe: increased density of a city could make being in that city more valuable, thus making more people want to live there at any given price, which means that prices would rise.
In a vacuum, no, I don't think simply adding housing can meaningfully drive up prices. But (as you say) obviously it feeds into organic urban growth, which does increase demand, so you have to keep building housing. It seems that the problem we have in our big tech hub cities is that we let that growth happen for a while and then decided that there would be no more housing.
I can imagine an induced demand effect as whole new classes of people realize that they can actually now live close to where they work, and come flooding back into the city. But like I said, that's an artifact of artificially constrained supply.
(I suppose increased supply could drive up price if providing that additional supply increased the overall costs but you'd presumably still be constrained by the overall supply/demand curves of the market.)
The burbs however....