Even if a large orchestra in rehearsal is not getting a verbal count-off, the meter is stil getting set up by the conductor visually--it's just that in many cases the conductor won't bother giving more than a single cue because the ensemble doesn't need it. Certainly if it's fast asymmetric and mixed meter music, you'll see something a bit more active from the director!
My perspectives on count-offs (or starting an ensemble with a visual cue) in general is that you are moving through the time immediately prior to the place where you're "starting". If you're at the beginning of a piece, your count-off is really in an invisible measure (measure 0, if you will). Some pieces start with pick-up beats in this otherwise "invisible" measure. So--whenever I am thinking about starting a piece (as a conductor or in rehearsal of chamber music), I begin my musical thinking in the measures or time prior to the "start point" and bring the ensemble in as we reach the appropriate point in time.
So long story short, in pieces with mixed meter I count off in the metric time that exists prior to where we're starting.
There's no reason to avoid these spots, and in many cases that's just not practical or doesn't make any sense (musical phrases don't always cycle in "nice" places).
I've seen professional ensemble directors do this in both of the ways that you mention, but obviously I agree with those that count in in the meter of the previous bar. I would never count in on an arbitrary time signature like 4/4 if it's nowhere to be found in the music you're playing--it makes no musical sense.
Lastly, I can see the 5/16 -> 7/8 transition being a bit tricky just due to the nature of the 5/16 bar (there's probably not a lot of time actually happening there)! In these cases, I see nothing wrong with extending your count-off to more than one bar; doing so in the meter(s) of those bars as they appear in the music.