If we're talking about the UK, then London is already as dense as it can reasonably be. It went all-in on public transit almost 200 years ago.
And of course, it made it all worse. Now you HAVE to work in London if you want a high salary.
The only real way to fix the housing is to promote remote work and decentralized companies.
"Reasonably" is doing a lot of work there. It sounds an awful like you're defining the boundaries of reasonability at "Exactly what is built right now and using the exact borders of current municipalities", which is a tautology. Even if we limit ourselves to the current municipal boundaries of London, population density is ~6000 people per square kilometer. Kowloon Walled City survived with 3,000,000 people per square kilometer, Manhattan has 75,000, Dhaka 23,000, San Francisco 19,000.
Their system is a much better one than we tend to have in the US, but "All-in on public transit" looks more like Trantor than London. A majority of the TFL system was constructed more than a century ago.
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Yeah.. no. Have you been to Asian cities? It's rows and rows of 100 floor apartments, it's urban centers consolidated into high density malls and commercial centers. We're talking about dozens of high rises where every floor contains restaurants and shops. I daresay there is more to eat and shop in the 1km Radius around Ikebukuro Station then entire borough of Westminster!
Nobody wants to work in some backwater in the middle of nowhere either, especially if you are young and want to meet new people.