How can someone think that? How can increase of supply increase prices? What's the logic behind that perception?

They see new builds hitting the market and look at the (aspirational) price put there by the builder. It's a borderline conspiracy that Zillow lists certain high-end new developments for a year (before they're built) which pollutes the "for sale" housing inventory. Whereas the "recently sold" is the best metric of how things actually are, the search options don't allow you to get an accurate market sense from these. After all, the app's purpose is not to give you an accurate sense of the market.

"New development" covers a wide range of development types. Most of the types that are "allowed" to be developed - per regulation, public investment, what does and doesn't trigger the NIMBY predatory instinct - are types that raise "market rates" like low-density SFH and luxury condos.* The government could "radically incentivize" developers to build high-density housing with basic finishes and fewer amenities, to be sold or rented at affordable rates, but that bumps up against https://news.ycombinator.com/item?id=36632171's impeccably correct observation: prices are high because the people who own the housing stock want them to be high. Inventory influences a fraction of the cost.

*(That is, however far you're convinced that there even is a market; for rent, thanks to "discounts" that allow property owners to avoid actually advertising lower rates, supply and demand don't exist, only the imperative to raise rates annually.)

Tokyo meets all of your listed criteria (though prohibiting on-street car storage is a big part of that) and they’ve been able to keep prices reasonable by allowing supply to meet demand. I think a lot of urbanists want American cities to grow in that direction: way more housing, huge transit investments, less subsidy for driving.