Why don't they let the seeker keep the snitch?

In Order of the Phoenix it is stated after the Gryfindor/Slytherin match that Harry had stolen the snitch. In The Deathly Hallows we learn that the snitch has a flesh memory to know who touched it first and is not reused.

Why don't they just let the seeker keep the snitch after they use it? Where do all the used snitches go?