Well, every mic you toss into the mix also adds more ambience, a bit of feedback from PA and monitors, and will pick up sounds that aren't really supposed to be heard at all, like breathing. Now, this isn't necessarily bad – in particular in the studio, I rather like the compact room sensation caused by many mics picking up bleed from different sources.
However, in a live situation you more typically can't avoid hearing too much of the (less than ideal, acoustically) room, you already have plenty enough background noise from the crowd to fight with, and you're close to getting feedback problems. All of these issues are exacerbated by many open mics on stage.
Thus, yes, the fewer open mics on stage the better is a pretty good guideline for live sound.
Nevertheless, I don't agree with Tim and would in fact support your husband's complaint. I used to always turn off unused channels too when mixing bands live, but don't do that anymore now, except for particularly troublesome sources (mostly quiet acoustic instruments).
In most cases, you will need to have all channels on at some point anyway, so you need to somehow make sure the mix sounds good even then. If anything is really close to feedback, then the sound will probably be destroyed by resonances already, so the only fix is to go back to the cause. Make sure the overall sound level is quiet enough, optimise the mic positions, tell the guitarist to turn his bleeding amp down a bit... tell the vocalist to make proper use of the mic, as well...
Such measures tend to have a much stronger effect on the final sound than turning down a few vocal mics, which, if placed cleverly, don't generally pick up that much unwanted sound. Overhead drum mics are more of an inevitable troublemaker in that regard.
As you say, any mic that's turned off is a risk of humilation for both singer and engineer, also a typical cause of overly hectic search for the problem, possibly dabbling with the wrong channel... it's just not worth it. Even if the engineer gets every mute right, that alone will keep them somewhat busy and distract from other things that might be more important to worry about. Better just pull the level of those channels down a couple dB, that usually already lets any interference become a non-issue yet if someone is determined to get through to the audience they'll be able to do it, albeit less than easily until the engineer gets the level right again.
A compromise that's getting ever more popular on these days' digital consoles is to just put noise gates on every channel. That certainly avoids breathing etc. to be heard in pauses, though it doesn't generally work that great for avoiding bleed (nor feedback, as Aaron remarks), because either you need to put the threshold so high that quietly sung or spoken parts get chopped up, or louder passages will have bleed from other instruments (particularly snare and guitars) trigger the gate open. Therefore I prefer to set the gates expansion ratio only to something low like 3:1, which gives a similar compromise to only slightly pulling down unused channels: unwanted noise will vanish in the mix, but important yet not-that-loud contributions will at least be audible at all.