Here's a real life experience with mmm's and bush mower at my property. Maybe something useful for your situation.
In 2001 we bought some property that included about 6 acres of "pasture", which meant it had once been a corn field that had been left untouched for 8-10 years. The first mowing, done before we bought it, pretty much worked the life out of the guy's tractor and bush hog. When I started mowing it, it was still knee-deep brambles, ivy and assorted weeds, and full of debris and rocks. I would hit it with the bush hog every month. By 2004, it was truly "pasture", and the natural grasses had driven out most of the weeds. In 2005 I bought a BX sized Kub with mmm to do small chores and go where the big Grand L would not go. Today, we can mow the place at 4" with the BX or bush hog at maybe 6" with the L. Usually we use the BX for the trimming and hog the rest. There are still some ruts that would be too much for a zero-turn mower, so it's not "lawn", but it's pretty close.
In your case, I'd probably start with a tractor/bush hog setup and use a riding mower for your 1 acre lawn. That would probably be most cost effective, and the bush hog will surprise you with how good your pasture will look after a few mowings. This also leaves your options open to upgrade the lawn mowing tools if you end up with a few more acres you want to keep finish mowed.
Also, hogging 10 acres with a 5' bush hog is going to take some time... probably an all-day job. You may find yourself wishing for a bigger setup to reduce the mowing time. Obviously a 6' bush hog would save you time, but puts you into a bigger tractor category.