My first job when I left school was at a Hunt yard we used to just flick over the saddle / bridle area before exercise, but when we came back we had to properly brush each horse, for about 15 mins with a leather body brush & metal curry come, you had to hold the body brush in your hand closest to the horses head and really put your shoulder in to it and get a good rhythm, I was fine with the brush in my right had but never quite got the hang of it in the left I had many a curry combed wrists! the idea is to get the blood pumping it also helps builds muscle tone if your doing it properly.And as I was told on more than one occasion if your not sweating your not doing it properly!
I agree that brushing off means a quick flick over before riding.
Regards strapping, that's quite an old fashioned term but to me it means a good all over groom and I do it EVERY day!