Very good calculator. I put in £10,000 deposit (not unreasonable for a couple yet to have children who both work full time to save £5k each) and can afford 1200 mortgage a month and going for the cheaper 2 bed places - children can share until they are 10+ as mine have.
That gives 78% of the UK affordable and just exclusives a bit around London.
Prices of entry level 1 - 3 bed places have risen in places where there are jobs on the whole. They have also risen because base rate is very very low and many mortgages only 3% interest rather than the 12% some of us used to pay. My mortgage was £90k a year interest a year at one point (not fun).
If you wanted no passing of mnoey between parent and child you would need to stop transfers during life times. I am giving the children what I can now in my 50s as I would much rather they had help now in their 20s then in 30 years' time when otherwise the state might take 40% in inheritance tax. There is no tax on life time gifts although we used to have it I think decades ago - capital transfer tax. If you could not make a gift to anyone without paying 40% iof the gift to the state that would be a bit complicated to enforce.
Also you could say those parents rich enough to have a spare bedroom could be penalised eg I don't charge my adult son rent so he is probably saving about £12,000 a year. You could change the law to say those who house adult children or their parents should pay 40% tax on a nominal rent from them even though no rent is paid. That would remove the unfairness of some parents housing children and others not.
You could also limit spending my parents on things like tutors for the children to get rid of that kind of advantage as that also tends to ensure the children get better paid jobs which is unfair on those children whose parents put no effort into them at all.
It is very hard to iron out disadvantage in practice.