Just finished my 18th year of teaching. During that time I've coached wrestling for 17 seasons, swimming now going on 13 seasons. Even helped with track for a couple springs and coached JV baseball one season my first year teaching.
In that time I've met / worked with 1 coach who backed up his talk with regards to kids playing multiple sports, 3 coaches who were adamantly opposed to "their" athletes doing something else and the VAST majority of coaches simply give lip service on the issue. That is to say when asked about their thoughts on the matter they are in favor of the concept. Do they do anything to encourage it? absolutely not. Do they get in the way and stop kids? absolutely not. They just say what people in interscholastic athletics want to hear.
I work with a veteran girls basketball coach who is one of our special edu. teachers. He "requires" his girls to play volleyball or at least run XC in the fall and then in the spring those same basketball girls do track. He attends their meets, even volunteers to help at meets. Backs up what he believes. We're the 5th school he's been at in his 30 plus year career and has been successful at each stop including ours.
We had a HFC years ago who would only let "his" football players do a second or third sport if he was the coach of it. So not only was he the HFC but was also the weightlifting coach in the winter and head boys track coach in the spring. He could tout that his football players were multi sport athletes but only because they did sports where he dictated the narrative. This HFC won 15 games in 5 seasons.
Not sure I've ever concluded that one sport is worse than the rest when it comes to this. IME I've noticed that it is difficult to get soccer players at my school to do anything else but soccer. This is mainly because soccer in South Carolina is played in the spring and a lot of club soccer is therefore done in the fall. Both our boys and girls soccer coaches don't force their kids to play fall soccer BUT I've never really heard them encourage players to do something else besides fall soccer.
Why does this happen?
Sometimes coaches who are teachers feel that the performance of their team on the field, court, etc... will boost the likelihood that they will continue to be able to work full time at the school they coach at. With regards to sports like football, basketball and in some schools maybe baseball this can be true. Your coaching performance is / can be tied in certain cases to whether or not you get retained as a teacher.
I've met some real egomaniacs in the coaching profession across all sports. Men and Women who were very successful coaches but honestly had zero business working with young people. Their coaching success was deeply tied to their mental well being and vision of themselves as who they are / were. In their minds to maintain their team's success they had to dictate everything about their athletes.
In the minds of some coaches they run the risk of that kid focusing on the other sport. They want their sport to be priority #1 and the second sport to be just sort of a hobby for the kid.