As it stands, it LOOKS like I can take (for instance) a PG and 'Start at C'... and he'd get the Center Training for that week.
You are correct, it's exactly like that.
Then it only showed 3 Centers with Minutes in that Week's Game... I figure the only 3 that actually played Center in the Game.
Again, you are correct. If you train one-position on C, there is only 3x48 minutes available to players. Since 24 minutes doesn't equal half of training, but less, it's strongly recommended that you don't split your minutes between your players, but have only 3 of them get 48 minutes. Which brings us to...
Strength in numbers.
There can be only 5 players at once on court, so you don't gain much by training 8 players at once. Actually you lose, because your centers will train slower than the ones that receive full 48 minutes. You can argue that you will create great backups, but backups play 12-15 minutes...do you really want to maintain two centers with same salary only because you want to have great backup?
Furthermore, only games in playoff and cup count. You can lose a bunch of games in regular season and come fourth, but then if you win the championship, who cares if you were 1st seed or 4th seed? It's true you had to have much better team to overcome home advantage of your opponents if you were 4th seed, but it can be done. So you don't need "great backups" to win every game; you need them for few crucial games/season and that's it. For those crucial games, you can train either RB for PF/C to let your two best centers play together, or have a team training.