Not trying to argue with you since you're obviously way more experienced than I am when it comes to playing and teaching - just trying to add a different perspective: Half a year is not that long, so what can you really expect when the focus is on drum set and not rudiments? To me it sounds like, the OP is progressing nicely and as the teacher is happy with the progress, I feel like this it the mind set that the OP should have as well.
@GCRoberts: Did you talk about this with your teacher? If you feel like you want to invest more time into rudiments and pad work, I'm sure your teacher would be happy to change course for a while since he proposed both ways to get you started. I also think that the rudiments you know already are a very good foundation for playing grooves, fills and short solos on the drum set. Keep at it, be patient with yourself and have some fun making music - You can do that without having perfect technique. Refining the basics is never "finished".
I think we agree here. But man, if a student wants to push harder, I’m not going to slow them down. That’s why I said “supplement”. Additional metronome practice is not going to hurt, especially if you’re using a mirror and really being honest with yourself about your form and stick heights. You can make some real progress in six months, especially if you don’t skip any days.
Also, it’s not unheard of for a music teacher to get a bit stuck in their ways. Could be a little of that going on here.