In many larger cities, older housing occupy prime location where the value of the land is twice the value of a new building you could build on it, so all these would never filter down to affordable levels.
Another factor contributing to the same thing is that cities aren't closed systems. Building new units at the top of the market doesn't mean that everyone in the city will move up one rung in quality leaving the cheapest unit now vacant. Many cities will have a latent demand for housing that is being suppressed by high prices. When that exists, someone from outside the city will move to the city to occupy the new construction or the newly vacated units at the top of the market. All that latent demand needs to be satisfied before the benefit of the new construction will trickle down to the bottom of the market.
But if we are talking city states, why not take the Singaporean approach? Just have the government build apartments and subsidize sale to citizens? Everyone gets a flat eventually.
IMO the HDB flats should not be saleable on the open market and can only be sold back to the government (subject to inflation and depreciation adjustments that no-one's going to agree to) but I'm an expat and not eligible to partake in the upside so I might be a bit biased.
All that being said they have done a better job than most of making government housing both accessible and appealing even if it means that I'm having to find a new place to live in the middle of an inflationary period (even as the country is seeing net emigration for the first time in decades, wth is going on??)
Only if you overthink it. An "open border" LVT+UBI system is self-balancing; a flood of immigrants would drive up demand for land, which drives up its value, which raises LVT revenue, which raises total UBI disbursement, which keeps each individual's UBI stable.
That said, it would work even better at a state level, and better still at a national/federal level.
> why not take the Singaporean approach?
The Singaporean approach (IIUC) includes land leases, which is in practice equivalent to a 100% LVT (assuming the government prices the lease at the land's full value).
Which is to say, what you've described is a signal to demolish and build higher intensity, while the filtering happens elsewhere.
For rental apartments, the most expensive ones are both renovated and in a prime location which is often still a centennial building. Brand new buildings are most of the time outside of these locations.