OK, so from that we can infer that between the 1970s and 2005-ish, mean household size fell significantly. That was a time when net migration was negligible as a proportion of UK population, so the drivers would have been a very slowly increasing population, a desire for smaller households, and housebuilding that apparently was able to keep up with that. Since 2005-ish mean household size has remained the same, despite what appears to be a continuing strong desire by people for smaller households, and house prices have rocketed. Meanwhile, the rate of housebuilding has remained stable since the early 1980s (
https://www.statista.com/statistics/746101/completion-of-new-dwellings-uk/).
The only plausible causal factor that changed between those periods (roughly 1980-2005 and 2005-today) was that net immigration rose sharply from around 2005 onwards, causing a much higher rate of population increase. So it doesn't seem unreasonable to conclude that it is in fact, population increase due to high net immigration (and not falling household sizes) that is that main cause of the current housing crisis and the unaffordable house prices.
Whilst it's a factor, and quite a significant, it's certainly not the main one. Yes there's currently about 10 million people who live in the UK who were born overseas, however a number greater than zero of these would always be able to come and live in the UK almost regardless of immigration policy (unless you are going to deny access to the UK of children of British parents), however even if they all came here in the last 20 years (unlikely) with a household size of 2.4 people you would have only needed to have built 209,000 homes a year.
That number of homes has generally been delivered each year for the last 20 years, overall maybe we are likely to be down by a little bit due to those years were we fell below that value, however generally not far off. Hardly a significant shortage though.
Although there's also a non zero number of people who already lived in the UK who were born overseas (4.6 million) and whilst it's impossible to be able to be sure of the numbers who are still living here from then it's certainly more than zero and do for every 100,000 still living here that's an extra 2,083 fewer homes needed each year (if all of them still lived here it would have been around 95,000 fewer homes a year, however that's unlikely).
The outcome is likely that to accommodate the number of new homes would have needed to have been somewhere between 130,000 and 175,000, which is comfortably below the number of homes being built, and so other factors do play a part.
Household size is a factor, (and it would have fallen further without migration), in 2000 the population was 59 million (it's now 68 million, so it's risen by about 9 million, so on a household size of 2.4 that's only 188,000 houses a year over the last 20 years, so we should have built enough to cover that rise), if you apply the UK household size in 2000 (2.38) and compare that to that if 2023 (2.36), whilst the numbers don't look that different when applied to 59 million it's the difference between 24.8 million homes and 25.0 million homes.
That's perhaps down to changes in who lives in their own, between 2000 (11.86%) and 2014 (11.66%) the percentage of people who lived alone had hardly changed, although this hides a significant shift upwards in age of those doing so, with those over 75 making up an extra 0.5% of those living alone, whilst those aged 16-24 falling from 3.45% to 1.94%.
Whilst an extra 20,000 homes a year doesn't sound a lot, when you add it to the 188,000 per year for the actual population growth of 9 million, you're back to 208,000 a year, so shouldn't be an issue.
However not all homes are created equally, and so if you've got an aging population where they are blocking up the larger homes (for example my parents live a 5 bed house) whilst those with children are more likely to live in smaller homes (for example, whilst we are fortunate that our kids each have their own room we don't have an office, even though my wife works from home an average of 2 days a week) you're going to create pressure on the housing market.
Likewise, there's an argument that second home ownership has also impacted on this too with 650,000 in 2008/09 rising to 809,000 2021/22, so in that 13 year window an extra 159,000 homes would have been needed so an extra 12,200 homes a year. Note this is truly homes people use as second homes and not all households who own more than one property, with the majority of the latter are renting out their additional properties.
There's also been a significant rise in the number of adults who own properties to rent, which further adds pressure on housing. It's harder to find year by year data on this, however between 2000 and 2017 it's been suggested that it rose from 3.6 million adults in 2000 to 5.2 million adults in 2014. That's at least (assuming 1 home per person) an average of 114,000 homes a year no longer available for people to buy.
Whilst those fingers are available to rent, so don't need "replacing" by building of new homes (maybe a small number due to the potential for small grapes between renters), it does reduce the ability for so many to be able to buy homes.
However, even by the 1990's there was already a gap in the number of homes being built, as compared to other European countries the UK should have built around 4 million extra homes over the last 50 years. To have kept up would have required an extra circa 80,000 new guinea a year.
Also, there's a big question to which the answer is unknown, how many homes would have been delivered if it wasn't for overseas workers?
At around the turn of the Millennium it was very costly (if not hard as well) to find a decent trades person. The influx of Eastern Europeans made this a much easier and cheaper thing to do. Whilst this resulted in a lot of ill feeling amongst those builders who saw their wages fall (which certainly had an impact on the Brexit debate) it also meant that it was easier to get more built.
It's a complex thing and whilst immigration has a sizable impact, we were already not delivering enough homes by 2000 for the aging population and then there's benefits in the ability to build the number of required homes.
Bringing it back to the current election, the Tories are saying we only need 200,000 per year, which means we're likely to continue to have a shortfall and not improving things.
Labour are keeping the 300,000 per year, so at least trying to keep house building at a level which should give us a chance to not get any worse and if the target is met could start to see things improving a little.
Lib Dems are wanting to build 400,000 per year, which is likely to be missed, however, if they could get the numbers up further than Labour then things could start to improve further.
One policy which could help would be something along the lines of the Labour New Towns policy, only with the land having been purchased by the government and there being thought through employment and public transport infrastructure plans to ensure that they don't become dormitory towns.