Would a new project idea that's similar to Teespring.com be successful?

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.

I think you're looking at this the wrong way. Your suggesting building a competitor for Teespring.com and focusing the features that Teespring doesn't have (or limits) with the assumption that those features are what is stopping Teespring's growth. What you should rather focus on is figuring out what is the most frustrating part of using a service like Teespring as seen by existing users and see if solving that problem has value. Building a pure copy with some added features is probably not a good idea.