My understanding is that a mortgage and home equity is not like equity in a corporation, where the bank initially owns some 80% of the house and you slowly buy the rest of it back a few percentage points at a time.

A mortgage is just a big loan with a decades-long repayment plan, with a special condition of the loan being a lien against the sale of the house and the right to foreclose on the house if the homeowner defaults.

That is the end of the bank's power regarding the home. The bank has no special power to enter your property or prevent you from modifying it, but also has no obligations for anything related to the house like liability or property taxes.

Is this correct?