The solution is pretty basic and was proven to work for centuries before extreme zoning intervention by governments happened at the behest of property owners wanting their gentrification.
To fund the program you institute luxury property taxes on units larger than what you want your average citizen to afford and actually property tax businesses (which many cities do not do, because they bribed companies to locate there with agreement they would have substantially reduced taxes). Both should be at progressive rates, the larger companies and homeowners paying more. In any city that has the modern housing price crisis in effect there is plenty of money floating around to pay this and would not "kill" the city to institute because demand is so high already.
Then, with the revenue, you A) invest in public and modernized infrastructure and B) rezone for mixed use or simply dezone your city (the only real valid arguments for zoning nowadays are to avoid overburdening infrastructure and to maintain specific skylines, but policies that prevent vertically entirely are wholly draconian and cannot stand - there needs to be an avenue for vertical growth no matter what).
The infrastructure is ridiculously expensive. Decomissioning streets to build functional bike and walkable space infrastructure, building a comprehensive subway / light rail / tram / trolly network, supporting comprehensive bus service are all major projects. Every city, though, needs to be doing all of these. Private car ownership has to trend towards 0 for anyone living in the city proper, and proper zoning and tax code should incentivize almost anyone within a substantial difference to move towards the city because its cheaper.
Its cheaper to live in a 50 story condo building when you factor in all externalized costs - the lifetime of the building, the heating / cooling efficiency, the infrastructure requirements, the construction efficiency, the materials efficiency, the cost of public services to support the building, the access to productive labor and jobs local to the residence, the climate damaged caused by long commutes from suburbs, the environmental impact of sprawl, the maintenance on infrastructure to support it. People look at how 2500 condos might cost 100 million for the building alone and think that means its inefficient and too expensive... but in practice A) the costs are inflated due to regulatory impedance B) there isn't enough high density demand in most western countries to get economies of scale reducing prices of construction the way they are in Asia and C) even when the building is more expensive everything else is cheaper.
Which probably comes back to the fundamental reason we are in this state in the first place. Externalities are not factored in. Environmental harm, infrastructure & services upkeep, human potential lost, etc all factor in to cost the modern sprawl low density lifestyle and our societies dearly. Cities are actively hurting their responsibility to grow to mitigate this damage, but its also on nations as a whole to stop subsidizing lifestyles that hurt the plant, nations, and people themselves. Cities don't have money to build because money is siphoned to maintain the status quo and that keeps this unnatural unsustainable relationship in place.