What is the harm in paying interest to the bank in exchange for a loan to buy a house, especially if that leads to owning the house after a period, just like paying rent which does not result in ownership?

The most dangerous thing to a Muslim's faith is corrupt analogical reasoning (qiyas fasid), which likens two different things that the Sharia has distinguished between. It is the root of every innovation (bid'ah) and corrupt statement. Analogizing usurious interest (riba) to a lawful lease (ijarah shari'ah) resembles the pre-Islamic era's (Jahiliyyah) analogizing of riba to trade, even though Allah Almighty permitted trade and prohibited riba with a definitive text. Riba is definitively prohibited, and analogizing its permissibility to a lease is a grave matter that impugns one's faith, because a lease is a contract of exchange, while a loan is a contract of benevolence and kindness, not intended for profit.