In the U.S., the mortgage holder (e.g., the bank) has NO ownership interest in the property. When someone buys property, they borrow money from a bank and then use that money (plus some of their own) to purchase the property. The property is then transferred to the new owner. When you get the mortgage, the bank is going to require that you give them a lien on the property. If you fail to pay the mortgage, the bank then has to go through the foreclosure process to take ownership of the property. Only then can they sell the property, and they are only allowed to keep as much money as necessary to pay off the outstanding debt, any extra goes back to the debtor.
The whole notion of "giving the property back to the bank" is incorrect, as is the idea that the bank can sell the property and make a profit.
The reason the bank wants a big down payment isn't so they get more money in a foreclosure and sale, it is to ensure that if they have to go through that process that will be able to sell the property quickly and for enough money to cover the debt. They also don't want to go through this process, so they want the borrower to be incentivized to make their payments (or sell the property themselves to pay off the debt).
The primary issue with self-employed borrowers is the bank needs to see consistent income for many months (typically 2 years). There has to be some assurance that the borrower will still be able to make the payments for the next 3-4 years (at which point, enough equity has been built up to mitigate most of the risk). Many sole proprietors tend to co-mingle their personal and business accounts (or simply don't keep business accounting records), or try to structure their income in such a way as to show little to no income (by writing off everything, typically for tax purposes). Unfortunately, you can't tell the IRS that you only make $10,000/yr and then try to tell a bank that you really make $100,000/yr. You also can't say I make $10,000/mo, if you can't demonstrate (with documentary evidence) that you actually receive (personally, as payroll income or cash taken out of the business, properly documented and taxed) this amount every month from a source that will continue at that rate indefinitely.