"The value of pure existence proofs consists precisely in that the individual construction is eliminated by them, and that many different constructions are subsumed under one fundamental idea so that only what is essential to the proof stands out clearly; brevity and economy of thought are the raison d'etre of existence proofs . . . . To prohibit existence statements . . . . is tantamount to relinquishing the science of mathematics altogether." - David Hilbert
I don't mind existence proofs because I figure at some future point, some way to construct the entities will be made. In some ways, existence proofs point the way towards clear research programs.