>lending institutions instead buy the asset (in this case, a house), and lease it out to the would-be borrower on an installment basis until a predetermined sum has been paid

If the would-be borrower misses a payment, do they get to keep a part of their total lease payments? When I take out a mortgage, the house is effectively owned by the bank and I also pay "installments until a predetermined sum has been paid". If I'm not able to pay the "installments" then I get kicked out.