When I was running a crew and selling, I would nly immidiatly agree to a removal if there were signs of decline in the plant. Otherwise I probe for reasons and try to speak for the tree (Advocati Di Arbor, Who knows the proper Latin form of that? I want to put it on my Biz Cards.).
"all it realy needs is some deadwooding, and directional pruning to keep it away from the house! We can do that for a quarter of what a removal would cost and you keep the shade. We get tthe dead out 3/4 inch and larger and you wont be doing anywhere nears the raking you are now! Okay we can get dead to 1/4 inch dia for $XXX. Sign here, please.