Don't forget politics. Support seem to be stronger to finance cooperative projects than national ones on scientific and exploration fields. And long term projects really need to have a good number of partners to become independent of short term political cycles.
I'm not thrilled with the ISS, but it is an accomplished of cooperation and the only manned space program that exists today.
In the times of competition, strangely enough, there were only Soviet/Russian space stations (and the short lived American Skylab). Even with the reference and stimulus you mention from competition, the Japanese, European and American station programs didn't materialize.
Yes, the Chinese have made progress and can be see as competition, but have only done one short EVA... I think that Intercosmos cosmonauts did more that that on the Salyuts

. And Intercosmos was cooperative.
You do have a point, but in my personal view, cooperation has lead to results within the current financial, technical and political global conditions for long term projects. Now, if we are talking about space tourism or private resupply missions, it's each man competing for itself of course!