Simply cutting off the limbs will badly unbalance the tree and so long as they're not rotten they'll still provide support to the branches cabled together. Tapping on the buttress roots and the trunks suggests the is mostly solid, with a decayed core.
Are there any accepted tests for stability (pulling on lines to the crown, or pushing on jacks applied to the trunk, for example) that can be used to gauge the stability of the tree as a whole? Obviously, if a tug on the crown tips the base of the trunk, it has to come out now. If a pull or push on the upper part of the tree doesn't disturb the root crown there's some hope of another year or two of shade.
A set of photos is at http://www.zefox.net/~bp/mulberry/ The images can be laid out something like
westlimb eastlimb crown trunks base to get an overall view if you're curious.
I had the cables installed in 2009 and the tree grew vigorously until the middle of this summer. Even now, the surviving limbs are well-leafed and seem healthy, thus my desire to save what I can as long as I can.
Thanks for reading