Yes, they do (or they buy the land and build new houses). I live in a rented house right now in northern germany. Most of the houses in my Siedlung are rented. My Landlord owns more than a few of them.

To buy a house around here you need upwards of 400k€ (there are cheaper houses, but then you will spend a TON of money into Sanierung), and for a credit of that size the bank wants an Eigenbeteiligung of around 50k€, I dont have that kind of money laying around, so renting it is. Luckily my rent is still kinda cheap at about 1200€/M.