"I think the only advantage in counting breaks is just that you're bringing attention to a break."

There is also - at least, at some non-beginner stages - the advantage of documenting the exact day-to-day progress in one's level of control.

"Crowley himself said that the number of breaks isn't particuarly as important as the continual effort itself to note the breaks."

I don't remember that quote (where is it?), but I would agree that the increase in mindfulness is important.

"In fact, as you get better you might be deceived that that you're having more breaks when in fact you're simply just more aware of more of them."

Yes. That's a definite advantage of documenting the number and type.

Probably the only thing I have to add is that progress is sometimes made better if, at the beginning of undertaking this practice, you not worry about tracking the breaks because that proves more cumbersome and discouraging. Get at least somewhat good at the practice before you start adding this extra layer.

This has the disadvantage that you don't get to see the enormous progress in the early stage from 3,000 breaks in 5 minutes down to only 100 or so ... but it does make it more manageable for a beginner. Once the numbers get small enough that you actually can spend more time meditating than counting, the break tracking seems far more valuable IMHO.