You're asking about professionals in one breath and home studios in the next. In my opinion those are two very different things.
so do most pros use the preamps on the board itself or patch in external pres for recording?
They probably use both. If you have a nice big neve or SSL or something, you're certainly gonna track through it.
If you've that much money to spend on a console though, you're probably gonna have a few other boutique preamps lying around.
Guys like that know their gear and they know which voice/preamp/mic combo is gonna work.
Any professional setups with analog console that I've seen are using consoles that cost about the same as my house.
OK, there are certainly cheaper quality mixers available, but there are also a lot of cheap mixers available that just aren't worth incorporating into a home recording setup.
I don't know what stage you're at but when I first started out I bought a behringer mixer because obviously it's not a studio without a mixer, right?
I learned very quickly that the functionality just wasn't needed and the quality wasn't worth putting in my chain.
I now use a motu interface and 8 outboard preamps.
Which is the best way for a home studio setup to mix?
My opinion is, the best way for a home studio user to mix is to buy a recording interface (or converters and separate preamps) and a computer, then just mix in the box.
Alternatively, use a standalone digital recorder.
Bottom line? It all depends on how you prefer to work and the quality of the mixer in question.