Yes, they do. On the 05 Elite team this year 3 were promoted from last years Second team. And, the year before that 2 were promoted who are 05's. There were others but those players were 04's.Nothing wrong with playing on the B team or C team. Before signing on the dotted line though, it's up to parents to make sure the level their kid is placed in is a good fit in terms of the coach, group of players, and level of play, and to educate themselves on what they will be getting for the coming season so it matches expectations.
It is true that it's rare but there are some clubs that do promote from within. LAGSB does.