Workers can own businesses today, so there's nothing stopping that from happening. There are lots of worker owned collectives out there. The major difference between Social Democracy and Democratic Socialism is that Democratic Socialism wants to nationalise many categories of business, ie. to have the government run them.

This tends to not go well where it's been tried because once business and government meld it becomes an unstoppable force that eats everything and has no accountability to anyone or oversight by anyone, and rapidly becomes inefficient and ineffective. I much prefer the Scandinavian way of mass deployment of labor unions, so that labor is a powerful force in it's own right rather than one that's ruled by a powerful merged political/business elite.