And don't forget all those off-mic moments - Jonathan Richman and Hamell on Trial are both acoustic artists where I've been more than glad to to have an audience tape due to the amount of stray from the microphones.
And sometimes practicalities win out: an extreme example - on Friday we taped a band that consisted of vocal, acoustic guitar, cello, double bass, drums, and keys. The board tape is great - probably the best I've ever captured myself - there's real warmth there and everything seems to gel together very nicely. BUT the band played an extraordinary encore, in the middle of the room, unamplified and, of course, there is nothing of it on that tape. If we hadn't have been running mics too we would have lost that amazing performance - the part of the gig that was probably the most memorable for those that attended.
I guess if I had to choose one or the other at a small acoustic gig, there may be significant temptation to go for the sbd but I know I would ideally want both board and mics if possible.