In short: interest rates & tax benefits.
Mortgage rates are typically far lower. My rate is about 1.5%. In Denmark it's -0.5%, yes you get paid to borrow.
If you look at mortgages, you have two negative cashflows: the principal you pay back, and the interest you pay on the loan. Of these two, the interest is actually the only cost. (taxes/insurance is separate because they're typically paid regardless of whether you have a mortgage, you'd still pay this even if you bought the home in cash without a mortgage.)
Paying back principal is not costing you anything. As in, suppose you had a $100 loan and you paid it back, you now have $100 less cash but also $100 less of a loan. You're not any richer or poorer. Similarly, if you do the opposite: borrow $100, sure you have $100 in cash, but you're not richer, because you still owe $100.
In that sense, if you borrow $250k for a home and over the years, earn $250k and use it to pay off the house, you've not 'lost' $250k. You still own it, it's just locked up in the home. If you sell the home, you get the $250k back, assuming the price hasn't changed.
But the interest, that's gone forever. That's where the real cost is in carrying a mortgage.
As such, the borrowing capacity and housing costs that someone can bear, is substantially influenced by mortgage rates. In the US these are 2-3% or so, in Denkmark it's negative 0.5% or so. These interest rates essentially dictate how favourable it is to buy, and how much more prices can rise, while still being affordable. Just compare the monthly fees of a 3% interest mortgage to one with a 1% interest mortgage. Or more informative even: compare how much you can borrow with say $1k a month in mortgage fees (principal+interest, or just interest whatever you want), when the interest rate is 1% or 3%.
For example, a $200k loan at 3% is $300k in total fees. A $280k loan with 0.5% interest is also $300k in total fees. And that's including principal. If you take out the principal, the difference is far greater.
On top of that there's significant tax benefits to home ownership in much of Europe, like deducting interest payments from your taxable income to a very large extent. The combined effect is that a lot of households have tons of debt which is actually very affordable and not problematic. (until interest rates will rise, which will get ugly... but most economists expect low rates to be the 'new normal'... we'll have to see. And a lot of the rates get fixed for 10-30 years so the impact if it happens, is not very direct).