That's a huge amount of tree hanging over that house in a codominant situation (which may or may not exhibit included bark - can't tell in video). And, that's a lot of decay at the base of the tree. Correct me if I'm wrong, but it looks like the decay is on the same side as the large leader growing over the house? I think anything other than removal is a gamble. If the homeowner is willing to take the risk, then kudos to them. I love large trees and that one is beautiful, but if that's my house and those are my kids sleeping in there, I'm cuttin' it into bits and I'm feelin bad the whole time I'm doin it. Call me uneducated, unsophisticated, or emotional. Here's my "professional diagnosis": that tree cannot be made unquestionably safe.
I agree, there is too much liability hanging over that house. No one can say exactly when a tree is going to fail. This tree as two red flags going against it, my vote is take it down.