In American sports vernacular, a trainer is not someone who helps the coach teach skills. At best he might train stamina and conditioning, but that person is probably called a strength and conditioning coach. At the junior level, the trainer hands out towels and carries the clipboard for the coach.
Whether the Celtics win the NBA Championship is probably not resting on Ed Lacerte's shoulders.
The function that is being performed by the "trainer" would probably be called an assistant coach, especially if the head coach is someone like Doug Moe who sits in his office handicapping greyhound races while the assistant coach runs the practice.
But then it would naturally occur to people playing a game like BB that there might be something like a head coach function, someone who does a better or worse job at motivating players and making substitutions.
The use of "trainer" is kind of like the word "practice" - it doesn't mean much to people, or means something entirely different. It helped changing "practice" to "game shape". It would make sense to change "trainer" to "assistant coach".
Then when people ask about the "head coach", it can be explained that they are the head coach (and GM and President)