Suppose there is a clock that determines the end of the game. Then Quidditch becomes similar to football (of the non-American variety) or basketball, with a slightly more hostile field. But it loses some drama from the Seeker no longer having a role.

I do think that multiple snitches is a good way to go, but video games provide the right role for them: power-ups. Then there's lots of room to get creative. Suppose there are, say, three snitches active at any time, in order to get their effect they have to be held (but they also attract the attention of the Bludgers, which has lots of potential for manipulation), and they all have a maximum duration of, say, two minutes, at which point the Snitch is depleted and is replaced by another one from the supply (and it reenters the supply to be available when the next one is depleted). Probably the seeker gets a faint glow of whatever color Snitch they're holding, both to make it more obvious to spectators and to make it easier for the other team to try to interrupt the Seeker's boosting of their team.

1. Raw points: a golden Snitch is worth 20 points if you manage to hold onto it the whole period.
2. Amplifying team members: a blue snitch increases the points scored through the Quaffle by your team, and a red snitch makes everyone on your team fly a bit faster. A green snitch makes the opposing side's goal hoops a bit larger, and an orange snitch makes your hoops smaller.
3. Replacing team members: a white snitch causes the Bludgers to avoid members of your team (both making Beaters unnecessary and less likely to help, since the Bludgers would avoid getting hit). A black snitch halves or eliminates points scored by the other team (giving your Keeper a break).
4. Meta: the team composition seems unforgiving to increasing or decreasing the number of players, but it might be reasonable to have one that lets you field an extra player or knock out one of their players (likely chosen by the opposing team captain), but here there are significant frictional costs for moving players on or off the team that might not be appropriate for something that only lasts two minutes.