Yes and no.
Chess.com sponsors people to make videos for their service, write articles and content, and produce advertising. They spend money, and they make money. They sponsor larger individuals in the chess world, and more people come to their service because of it.
Where do you think all that advertising revenue goes? On providing a better service? Heh, no.
It's not half as expensive as you think to supply a functioning chess website. Lichess is primarily funded by a sole individual, Thibault; and monetarily speaking he's a pretty average person.
So no, you don't need costly memberships and invasive advertising to supply a decent chess website. No, nobody's really breaking their back to keep lichess running. In all honesty, it's just a lot of fun.
The people that stream videos playing on lichess, they do it because they want to. The people who make videos on YouTube: they do it because they want to. Servers have been donated to us. Code and media content has been sourced from the community.
So what you need is a handful of people who are dedicated to the idea of creating a powerful community based chess website. Not large sums of money.