Taken to the furthest conclusion, the home-owning class would eventually trade houses only amongst themselves, which leads to a need for a housing currency or trading pool to make trade easier.
2. Right now there's also the problem of homeowners not wanting to sell because that would mean giving up their low rates. This hampers supply and makes it harder for first time buyers to make it into the "home-owning class".
So to resell an assumed mortgage, you have to find a buyer that can pay the entire unmortgaged principle (i.e. down payment + already paid principle) plus any increase in value. If that buyer is another home owner with a low rate mortgage, they may have to find another buyer like themselves to assume their current mortgage.
The assumption itself likely requires the new buyer to be as qualified as the current owner as well.
So this is maybe attractive right now, where there are many of these low rate mortgages with high loan to value ratios, but it will get progressively worse. Add that on top of the friction and reduced addressable buyer market, and my intuition is that this is probably a wash at best.