That's definitely a viable, and sometimes simpler, approach. I've done both, at different times. There are a few reasons some mixers prefer to have the console scene trigger QLab, rather than vice versa:
1) There are usually on a musical more console scenes than QLab scenes, so it's simply less programming to do.
2) Some mixers find it faster to navigate scenes in the console than have to reach for the mouse and navigate the cue list in QLab when jumping around in rehearsal. This way, no matter where you are in the console, you always are automatically triggering the right cues in QLab. This matters less during the run of the show, but can be a nice efficiency thing in rehearsals/production. Where the difference really shows up is when repeating a sequence a number of times in a row in rehearsal. You can just back the console up those few scenes, and when you hit the point that QLab needs to be triggered again, it triggers. No need to navigate the QLab cue list to move the playhead.
It's one set of small tradeoffs for another, and it may be that, in the case of this particular console, having QLab be the one driving all recalls is a lot simpler.