This always has been a tough issue both as a parent and a coach (when I did).
As a single parent of two girls, it was already tough to send them to one sport with both playing on different age brackets can't imagine what multi-sport would be like. Financially, I couldn't swing it. If I did swing it, I may not have allowed it anyway. The reason, one team always loses. Missing a practice for another practice, miss a practice for a game or vice-versa. It affects both (or three) sports.
As a coach, I hated it. I had to carry 18 girls to play fall softball (granted it was only rec), and only half the time I was able to field a team. When coaching travel, it was equally as tough when girls were missing very important practices for rec games. But the girls were 9/10U. If I do field another team, I may put in a contract for multi-sport athletes the amount of time they can miss. I still have a few years before I do anything like that as I won't do it until my youngest is off on her own.
Most people say it good for the muscles as you can work other groups, sure from a resistance standpoint. But if you're a right handed in softball, you're using the exact same muscles to serve a volley ball, or pass a basketball. During the course of a year, those muscles will eventually break down. With the specialization of one sport, rest is a need now. But what do I know?