You are correct about the cost of storage. However that is not the cost of storing these videos. When you set up a server you just buy a hard disk and some other hardware correct. But Youtube also has to buy staff to maintain, update and services and networks, offices for them, legal stuff, free food, insurance, electricity and every other thing a huge business needs of course.
Google after 2007 has been about doing things that make business sense. Just storing these videos is a cost. It's why there's been a glacial pace in new features, APIs and support in youtube for over a decade - it's not a financial priority.