I tried to think of ways that we could get 60 yard times for gymnasts, and the best I could come up with is the Superstars, which were made for TV competitions among well known athletes, shown in the Seventies and Eighties. The idea was to take the best (and most famous) practitioners of various sports and have them compete in a variety of events. Generally the contestants could choose 5 out of 7 possible events, allowing them to skip events for which they were poorly suited. This setup was probably put in place following the near drowning of the boxer Joe Frazier, who later admitted that he had suspected he couldn't swim, but wasn't sure because he had never tried.
One of the events was the 100 yard dash. I remembered that a gymnast had been included, and so I hoped to get a 100 yard time.
As it turns out, in 1981 Kurt Thomas was included, finished second in his preliminary group, but declined to take part in the finals. Later Bart Connor competed, but did not fare well in the preliminary competition. (Thomas had the reputation of being a better athlete than Connor).
Unfortunately, both skipped the 100 yard dash, and so there isn't a dash time. I do think, however, that we can draw an inference from their choice to avoid the 100 yard dash: they didn't think they would do well.
So I think we can guess that world class gymnasts aren't competitive with other world class athletes in sprints. If so, the assertion that one "could pick any male gymnast and he could outrun just about every ballplayer in the 60 yd dash" is unlikely to be true.
On the other hand, I support the notion that gymnasts are typically in better shape than baseball players.