A lot of this is kinda already happening.

Score could be kept with citations.

This is already something people brag about / look at as a measure of success. There are plenty of free websites to keep track but the most popular one is Google Scholar.

Perhaps competing labs could both receive citation credit if their results essentially showed the same thing.

When I find multiple good papers that have the information I need, I cite all of them, and even feel happy about it because citing a lot of papers can make your paper look like you put in more work.

If nobody could scoop anyone else's work, then cooperation may be encouraged over competition.

It’s a bit hard to completely do away with scooping. A possibly more practical way to increase cooperation would be to eliminate the idea of the “first author” getting the majority of the credit. It’s really annoying when like 5 people heavily contributed to the paper but whoever’s name is listed first ends up getting 90% of the credit because that’s what people look for.

The idea of doing things in a wiki format is interesting though.