Certainly depends on your definition of sports, but...
Are there sports that are uniquely cooperative? YES
I can think of several examples where a team accomplished a goal of sport that was not exclusively against an opponent.
Edmund Hillary and Tenzing Norgay were the first confirmed climbers of Mt. Everest in 1953
Any time a team climbs a mountain, circumnavigates the globe or rows across a body of water - this is a cooperative sport. If you remove the constraint of first or fastest, it becomes non-competitive. (Conversely you could make a game like pandemic competitive by having teams see who can complete it fastest)
For a silly example:
Highest trampoline bounce (team) - You could argue they are competing against the previous record holders, but I'd argue they are competing against gravity.
Are there very popular cooperative sports today? YES
The ones mentioned earlier, but any action sport where teams work together to accomplish a goal - Think endurance sports, group activities that conquer nature, or other team building exercises like Ropes Courses.
Were cooperative sports more popular at some point in our history? HARD TO SAY DEFINITIVELY
Because this is not an established category of sport, it's hard to say that one time period showed more popularity - we can probably say that these types of activities where there is no competition require both time and money - as such these activities were probably limited to the ultra-rich in the past or those that funded such feats. Today people have considerably more leisure time. According to The Federal Reserve Bank of St. Louis, "Americans appear to be working less. Numerous economic studies suggest that the number of hours that the average American works in a year has fallen by about 550 hours from 1900 to 2005." This suggests that modern people have more time to partake in such activities.