Well everytime somebody wants to build high density housing or really anything to alleviate the housing crisis we get the usual NIMBY screaming similar to OP about evil corporations coming to destroy the character of their local neighborhood.
Returning by right development to the UK is very possibly the single largest policy measure that might enable a way forward, not because it's so intimately tied to the financialization of the economy, but because it's such an enormous capital concentration that its limitations overshadow a lot of other issues.
It may be hard to see it from where we're standing, but the current housing situation is one extreme of a catastrophic ongoing crisis.
"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.
Nope. London is reasonably full already. The problem is that it's already 9 million people, and the high-paying jobs are focused in a fairly small area.
So if you increase the population density, the transit to that focused area will have to carry more people. And it's already at capacity.
If you want to increase the density, then London will have to create new business areas outside of Canary Wharf, City, and Westminster. At which point, the question becomes "why even bother with London?"
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.