Small size is definitely one major factor, but it doesn't totally explain it, for 2 reasons:
1. These days we see more and more musicals that seek out more intimate houses, either because because the material calls for more intimacy, and a physically smaller production, or because small capacity increases demand. So why not the Golden, if the production fits?
2. The Golden isn't even that small, compared to some other theatres that have also housed musicals recently: the Booth and the Hayes are both smaller than the Golden, and both of them have housed musicals more recently than Avenue Q. The American Airlines and the Friedman are also smaller than the Golden, and both of them have also housed musicals in recent years (though it's much more rare for the Friedman) - but since they're run by non-profits, they wouldn't be as concerned with selling more tickets. Also, the Circle in the Square has only 35 more seats than the Golden, and they house musicals all the time.
If I had to guess what the other missing factor is here, I would guess that it's just a question of owner preference, and producer demand. Scott Rudin seems to enjoy booking it for his smaller productions, and it seems that other producers like to, as well. So the theatre is probably in high demand among producers who are looking for a small house to do a limited run plays. And because demand is high, the Shuberts might not feel the need to stick a musical in there. And maybe they feel that it's a valuable business tactic to keep the Golden from being booked for long periods of time (as most musicals are open runs), so they can maintain a fruitful business relationship with producers like Rudin, who need to the space to produce small, limited run plays.
This is all just speculation. I have no idea if this is how the Shuberts think. It just seems like a logical explanation to me.