There are countless competitors already to Teespring, what they do is not new nor did they invent a new way of printing for fundraisers.

I think what you're suggesting will actually hinder you. The finite time limit creates scarcity and exclusivity. If you are a member of a group or organization that wants to produce wares as a fundraiser, it is in your best interest to have a time limit. Having this open-ended will result in a lot of people putting it off until "they have time" -- which is never.

With this approach, you are intentionally undercutting your own growth.

Your best bet would be to capitalize on #2 and beat everyone else in the speed of service or find other services (like Teespring) to license your process to.