There's no guarantee a "single, unified global effort" will be better than multiple different efforts though, plus it sounds like putting all your eggs in one basket.
If events in 2016 has shown anything, it is that people are naturally divisive. We all live in our own bubbles and want to do things our way all the time, win all the time and be right all the time. Unless you're willing to compromise, it'll be hard to just tell people to team up with you and do things "the global way".
There's no guarantee they won't interfere with and nullify each other. For instance, one team lands a big ion thruster on the rocket to push it out of the way. The next detonates a nuke next to it. They screwed up the positioning so it didn't deflect the asteroid enough, but did wreck the ion engine. Oops.
At the very least you need some communication between the "competing teams".
If events in 2016 has shown anything, it is that people are naturally divisive. We all live in our own bubbles and want to do things our way all the time, win all the time and be right all the time. Unless you're willing to compromise, it'll be hard to just tell people to team up with you and do things "the global way".