This seems pretty strong to me
8.4 million is a LOT of people.
That also matches my personal experience - I've moved around a fair bit, and experienced life renting, buying and letting. My experience is that up until the late 1990's, if I wanted somewhere to rent, it was pretty easy to find places. Around 2000, it suddenly became very hard, and over the following 10 years became increasingly harder, with quite a lot of people chasing any suitable place, and rooms/flats/houses being snapped up very quickly. On the other hand, experience letting (in NW England) is that finding tenants is usually very easy.
House and rental prices and the % of income people tend to spend on rent also provides evidence - because houses do obey the normal laws of supply and demand, and rising prices therefore indicates rising demand relative to supply - although I appreciate care needs to be taken in interpreting this because of things like legal constraints on mortgage lending and buy-to-let, which will impact the market.
Money chasing homes as investments doesn't change the total supply of housing. Each time a buy-to-let investor buys a house, the reduces by 1 the supply of houses to sell, and increases by 1 the supply of houses to rent. All other things being equal, in a functioning competitive market, that would slightly increase market prices for buying, and slightly reduce market rents. Homes as investments therefore seems unlikely to be able to explain that rents have increased sharply over the last 20 years - whereas not-enough-houses explains that perfectly.
As for how many I'd want to build - that would take some analysis of demand, but I'd want to build as many as are necessary to ensure that in all parts of the country, there are as many homes as there are families+couples+single people who realistically need a home. Remember, back until the mid 1970s we were building about 300K homes a year, so building more houses than we currently do does appear to be possible.