we had this same problem rocky - when we moved into our bungalow 3 years ago, the garden had 4 large (and I mean large) trees in it. Mr. Den cut them down one by one and dug out the last stump last summer. He just cut through the remaining roots that went into next doors garden. We have'nt had any problems with regrowth etc.
I think what you are doing is the way to go, unless you get tree surgeons in and they cost an arm and a leg !!
One of our stumps was so big, it took Mr. Den and 5 neighbours to lift it out of the hole.