I agree with your conclusion, but part of your logic is faulty. You seem to be assuming that all savings takes the form of holding debt from others, e.g., bonds. But that's only one possibility. You can also save by owning equity in others, e.g., stocks.

However, I don't want somebody else to own equity in my house, rather me having a normal mortgage.

At the very least, such a change would cause all kinds of liquidity problems, and would certainly cause the significant negative effects you predict.