But it is likely to be the main factor. As I pointed out in post #105, the % increase in the population over the last few decades is massively greater than the % decrease in household size.
By the way, another factor that isn't often mentioned may be that people today want bigger houses/more rooms for the same numbers of people: For example, siblings sharing bedrooms - which was once normal - would generally be seen as undesirable today. And 50 years ago, very few people would have needed a home-office room.
Whist immigration is the main factor, needing to build 100,000 homes a year makes it quite a noticeable amount of the 300,000 government target.
The one thing which is harder to split out is how much immigration has done to stabilise the size of households. If for instance, the number of households with 2+ children had fallen further than the current number due to the lack of immigration increasing the number of children being born here. Then it could well be that households sizes would have fallen further.
The other thing to highlight, the cost of home ownership for young people is high. If it was more in keeping with where it was 30 years ago then it's likely that household sizes would have been smaller still.
In 1971 there were 55.9 million people living in 19.27 million homes.
In 2022 there's 66.66 million people living in 28.25 million homes.
That's best part of 10 million extra homes over a 30 year window, which is broadly 333,000 homes a year, so even the government's target of 300,000 falls short of this average.
Based on the fact that the target hasn't been hit for the last decade, maybe for the next decade the target should be closer to 500,000 per year.
To meet that target a place with 10,000 people would need to build an average of 75 new homes every year for the next decade (750 homes).
However, that's going to be a hard sell (and rather tricky everywhere, for example Tower Hamlets), which is why it's you're aiming for 5 million in the next 10 years I can see the logic of building new towns (5 with half a million in each could be easier than a lot of smaller developments).
I would also suggest that you look to build fairly high density, but with lots of green spaces. Rather than trying to build suburban sprawl, with all the ineffectiveness which come with it.
If you do this in a new city, you'll be able to do it. A village with 5,000 people in, would end up with their 375 new homes taking up far more land than the same number of dwellings in the city.
What's needed is a way of stacking dwellings with decent outside private spaces. This may result in a couple of flats at ground level (with a garden) and then a town house and garden sitting on top of these flats.
In doing so you could cater for families in the town houses and retired people in the flats (rather than needing a bungalow). As the cost of land is much reduced for each property (say the equivalent of two bungalows and a town house in the space of a pair of semis), it's likely that the cost of the properties wouldn't need to increase for the extra build costs.
Infrastructure costs would reduce (shorter lengths of roads and drainage runs) as would the running costs of the homes (due to more shared walls and floors/ceilings).
You'd probably orientate the town house to overlook it's garden (nearest the street) and the "bungalows" and their gardens to face away from the street to minimise the amount of overlooking. As whilst they would mean "front" gardens for the family homes, they would be at first floor level (so no overlooking from the street) with a fence around (so no overlooking from first floor level and depending on the angles fairly limited overlooking from the second floor of the houses opposite). As such probably no worse than most people who live in a lot of houses in suburbs.