if we are nit picking definition then to quote wikipedia:
> Capitalism is an economic system based on the private ownership of the means of production and its use for the purpose of obtaining profit
this neither implies free markets nor that "rich people control everything" but it more like a large category of economical systems we haven't always given distinct names
but common reasoning why capitalism and democracy work well together are based on a form of capitalism which fundamental requires fair and free markets.
This are the forms of capitalism the US had been fighting for in the past, the forms of capitalism most EU countries have committed to, but definitely not the form of capitalism we have in the US today.
The forms of capitalism which endanger instead of complement democracy have IMHO no place in democratic countries and IMHO we should give this harmful sub-category of capitalism it's own name to be able to better fight it.
Anyway just because there are bad forms of capitalism doesn't mean there aren't also good forms. And when pushing for "fixing" capitalism you at least have some chance to reach people. When you instead push to "replace" capitalism you are pretty fast hitting a wall where people aren't even willing to entertain reasoning with you (which is probably why the right wing tries to re-lable most things they don't like as anti-capitalist, communist or socialist even when the arguments have nothing to do with any of that (well the term socialism has been redefine and appropriated so often it has lost any meaning in taken out of context, but that is a different topic).
That's the same system where rich people control everything. There's no system where people privately control the means of production and none of them ever use it to expand their control of the means of production.