There need to be tighter legal controls on buying homes and leaving them empty. If homes are not being sold, the prices need to come down. But the UK has a low vacancy rate by international standards. There is no way out of this mess that does not involve a ton of building too, even if other measures need to be taken as well - the population has increased, people are living longer and there are larger numbers of households due to more singles, later marriage and so on.
The UK has also been failing to rebuild its old housing stock, which is why the UK has Europe's oldest AND heat-leakiest housing stock, AND the least floor-space per person; things like 19th century terraces are cramped and impossible to insulate properly, so people are cold in the winter and hot in the summer. Most housing only has a lifespan of about 150-200 years, and a lot of Victorian terraced housing was thrown up quickly in the expectation it would be demolished within a generation; our 19th century forbears would be surprised if they could time travel to 2024 and discover that modern people are still living in these things. Given that a lot of this housing stock will need to be demolished in the near future anyway, it might be a good idea to replace a lot of it with taller buildings (a mixture of three- and four-storey terraces and mid-rise apartment blocks would probably be a suitable mix for these kinds of areas).
Millions of people are on low wages (and someone has to do those important jobs). Plus there's disabled people unable to work, and full-time carers. They will never be able to afford buy regardless of how many non council homes are built.
The point of my post was "Let's do more private sector building as well," not "let's not bother with council housing." A certain % of the population will never be able to manage private renting/home ownership for various reasons, which is why council housing is essential. Yes, we should build more of that as well. We should be doing all the things, because the UK needs about 4.5 million more homes in total. "Building more" is not the only thing that needs to be done, but there is no solution that does not involve a ton of building of various kinds.