I'm not sure why people have this notion that we can only take the good parts of capitalism and leave out the bad parts. It seems this is what you're suggesting--but how would we do that? Because when you do, you're most likely making the economy _less_ capitalist, and more of a mixed market economy.
Seems that what you call a mixed market economy is exactly what I meant. It worked for a long time for quite a few countries here in Europe, until "free market" was suddenly considered a panacea for everything according to the ones in power. Before we got to the casino-economy of today...
That said, a free market is capitalism at its purest, and in a free market, we'll get sweatshops and the like (which are very sought-after jobs in developing countries).
I guess my point is: semantics matter. If we want to push back against these effects, I think it's important to realize we're actually pushing ourselves away from capitalism and more towards a mixed market economy, which is a good thing in my judgement. But we shouldn't see a mixed market economy and call it capitalism, because they're not the same thing--mixed market economies attempt to mitigate capitalism's negative effects.