It seems to me that, historically, it is simply adversity in a general sense (be that other humans or not) that has driven humanity forward. "Competition" per se is not the motivating factor, it is just that there is a problem that needs to be solved. When the necessity of problem solving diminishes, so does progress.

The competition theory of social organisation tries to stimulate this by creating artificial adversity through (typically non-violent) competition between people. I speculate that one of the key issues with this approach is that a lot of the problems that you create are not bounded by "underlying reality" as such - they emerge from the competition itself, and so we end up spending a lot of effort solving meta-problems which, if you stripped away the competition, aren't really "helpful".

Further, it seems difficult to distinguish which problems are "real" problems, and which aren't, because the competition does exist - what we end up with is a pervasive sense that a lot of what we're doing isn't useful, but we can't work out what or why.