The biggest quick win is open up specific category of build to rent 1-2 bed flats aimed at those at university or within 4 years of university/college, with rent caps at no more than quarter of average wages. Short term rentals, and regardless of how many different locations used, automatic eviction 4 years after graduating when no longer entitled to that category
The simple reason is many developments are bogged down in planning because of lack of doctors and lack of school places, and this group has negligible needs for either. There is also a huge pent up demand as young people often move around bit during first few years of working, but not in a position to commit to same location long term. The rent cap also allows chance to save for a deposit.
To give idea of scale, say 400,000 graduates per year, plus another 200,000 doing teacher training, nursing training, and various other jobs with further on the job training. Could comfortably fill 1.5-2m spaces. Whats more there are long term funds (eg pension funds) that want to invest in long term income generating schemes.
Before anyone says age specific property is wrong, we already have housing schemes where minimum ages of 50 or 55 or 60 apply.
The main advantage of going for quick win, is it indirectly starts to free up other housing. We (UK) currently spends billions on subsidising families to rent private homes, some of which are dreadful quality and privately owned. The reason we do this is due to lack of public housing (at end of 1970s before sell off started, was about 6.5m public (council) homes).
The other very fast win is to stop sale of council houses (and it's quasi equivalent housing association). There will always be reluctance to invest if got to give it away at a discount few years later. Perhaps more palatable would be only houses over 50 years old can be sold.
The experts advise not really a shortage of family homes, just lot of them are currently being used by non-families. Two main categories: split into multiple rentals, a category that can be controlled as there are licences for HMOs (houses of multiple occupation) which need renewing at intervals; and older people still living in big family houses due to lack of suitable retirement homes.
You don't need to be a genius to work out family homes are attractive and profitable for builders, so they will keep building more if we continue with turning them over to HMOs and encouraging older people to stay in them. In simple terms it is often seen as cheaper and easier for the retirees to employ a cleaner and gardener than spend thousands in one go moving to suitable retirement home. That's if they can even find a suitable retirement estate, they simply don't exist in most towns (it's often small flats in retirement apartment blocks, or nothing), no intermediate downsize homes.
The other area where UK is poor is co-building, self build estates, there is lot of brownfield land, and a few pilot schemes eg Graven Hill, Bicester, but negligible opportunities to build your own home in majority of towns. In theory legally councils are supposed to have list of available self build plots to peruse, but I bet getting hold of your local one is as easy as looking for needle in a haystack. We need lot more of these build own house approach.
So for all good intentions (and silly short term policies subsidising things like heating unsuitable homes), we have no strategy. The two categories desperately needed (early career rentals, and intermediate downsize retirement homes) are virtually never built, so family homes get tied up instead.