Not liking "Manhattanization" is fairly common in continental Europe as well, but it mostly works out because we build lots of housing on transit lines, so you don't have to dump everything into huge condo towers right in the city center. As for why? Part of it is that it's not really
Manhattan people are worried about turning into (at least that was built over many decades and has varied and interesting architecture), but
Miami. Or the Spanish coast. Older buildings built during various eras, with varying architecture/history, and a "human scale" that doesn't block the light/sky, are more popular than tearing it all down and replacing everything with a rash of Miami-style high-rise condo towers. It probably doesn't help that there was a
little openness to "modern high-rises" in the 1960s-70s, which led to some really terrible-looking monstrosities that have aged even less well than they first looked, now standing as monuments to "welp, let's never let the developers do
that again".
However, you can still fit quite a lot of people into 5-story buildings if you build them densely over a region and connect them well with transit. I don't think the problem with the SF Bay Area is that it has 5-story buildings, but that for the most part it doesn't have 5-story buildings, rather 1-2 story houses with yards. The urbanized part of SF is tiny, effectively a small part of the eastern side of the city around downtown, SoMa, and the Mission. Not even the Sunset or Burlingame are built up with 5-story buildings. And Palo Alto, despite being both a university/research and business hub that has housing demand sufficient to support much denser housing, is instead occupied by things that look like countryside vacation homes, failing to even approach "medium-sized town" level of density, let alone some wild dream of getting near the density of 5-story apartment buildings.