As for putting a deer who is starving out of its misery and offering this as a moral rational seems odd. Would you do this service for all animals who are starving: squirrels, badgers, frogs, snakes, raccoons, etc. Why do only deer warrant your compassion?
It's worth noting that all animals (including human animals) die of starvation, disease, or accident. Why you'd single out deer for special consideration to be euthanized by you for compassion makes little sense, except to defend the indefensible.
As for over-population, the only animal on this planet that is over-populated is human beings. Our notions of animals being over-populated have no ecological basis; such notions are entirely social. There are not too many deer ecologically, there are simply more deer than we would like. The use of the term "over-population" is mere propaganda.