No, he trains players, individually or in groups, or the entire team at one time. He is training one or more players, not one or more "positions."
Yes, and players play positions so I dunno what the problem is.
The players wear uniforms. By your reasoning, they might as well train uniforms.
Players play many positions, and the "positions" are highly fluid. A real coach still trains players, not positions. He doesn't say, "Tomorrow I am training shooting guards. Anyone who wants to train as a shooting guard tomorrow show up at 10 AM." He trains players, people, individuals. Training "positions" has no logic either for the game or for RL.
Training in this game is a lot like a stone in your shoe. You can have it there so long that you become accustomed to it, even grow a callous and barely notice it. You eventually become indifferent to it. But it is still nothing better than a stone in your shoe.