The snitch manufacturer wears gloves so as not to handle it directly. Scrimgeour brings the first snitch that Harry ever caught because he hopes to use its flesh memory for it to identify Harry. This implies the same snitch cannot be used between games since it will probably remember the previous person who caught it.
- Is it necessary to use a completely new snitch for every game of Quidditch?
- Can we erase the previous memory of a snitch for the next game?
- If the answer to 1 is yes, did Dumbledore really hold on to the first snitch Harry ever caught for all those years just to use it as a box for the Resurrection Stone which he did not even know if it existed at the time?
- If the answer to 2 is yes, it is probably something Madam Hooch would do, may be after the current game ended or before the start of the next game. Why would she hold on to the snitch and not erase its memory?