Nowadays that's not necessarily the case, as most will be freelancers specialising in whatever aspect of sound recording/mixing they're hired for...
However, you definitely have a point about the past! I worked with studio sound people a lot in the days of live Top Of The Pops; their thinking was the sound must be "as is", no FX, no making it sound any better as that was cheating and not putting out exactly what was put in. This, I might add was also when these folks were exclusively recruited as engineers - there was no time for fancy frills and expression, twas purely a matter of keeping it within spec and watching the meters. Their thinking was "if the band don't sound good enough on their own it's not our job to help 'em!". As a result, most bands sounded terrible! I also had them working for me on a well known drama, and the amount of grief we had just convincing them to actually add to the plot with a spot of intelligent sound design was amazing. Not all of 'em, I might add!
So I kept the best 2 or 3 and brought in sound dubbing mixers from Post Production (the others were studios) - their job was to make it sound good so that's what they attempted to do but because of studios demarcation, those mixers never got a look-in at music mixing.