I cannot speak for the intended purpose of ArXiv by its creators but I can tell you that, in the conference circles, its main intended use was flag planting: people were afraid that their competitors would tweet some results while their (earlier) paper was under anonymous review, and so researchers started putting their stuff on ArXiv first to ensure no one would "steal" their claim of being there first.
I don't think it has much to do with tweets, but the flag planting is correct. You have to timestamp your idea because otherwise if you're too slow, the community doesn't much care about "pretty much same idea but now done properly and thoroughly" and will just elevate the faster one.
If the researchers didn't care about promoting themselves they would not care whether a competitor published a result first or they did. Either way science is being pushed forward.