Do you think the insurance
company who writes my homeowner's policy might pay to have the tree
taken down or at least the dead limbs removed rather than eventually pay not
only for cleaning up fallen limbs, but also paying for any damage done to
neighbor's property?


Nice try.

They not only will not pay - they won't likely pay out a claim if it does fall
down (or will collect it back from you after having to pay your neighbour)
because you had a pre-existing dangerous condition that you knew about but did
nothing to correct.

Your tree=your problem. Try throwing rocks at your own windshield and see if
they pay for the repair work ;-)

You can't just sit on your hands if you have a problem that may affect other
people and say "Oh my - I guess someone esle will pay" - although far too many
people do seem to adopt your approach.