I think plenty of people do. You just don't hear about them because their marketing falls short, or is targeted to a niche market you don't pay attention to.
You can build an experience pretty close to WordPress.com with freely available code:
Plus some cloud infrastructure provider like AWS or Google Cloud Engine.
Easier said than done, but these days the technology isn't as far out of reach as it once was.
Competition is pretty ferocious for general website builders though. There are some lesser known ones like Strikingly and TheGrid, but I believe the best way to break through (albeit on a smaller scale) is to focus on a niche.
Examples include HappyTables for restaurants, Hotel Propeller for hotels, Dunked for portfolio sites.
SquareSpace is a proprietary hosted solution, by the way. Not WordPress.