Since you actually presented an intelligently-reasoned rationale for a "players rather than positions" methodology, I actually feel like this response won't be going to a brick wall.
The first point I'd argue is that "just 48 minutes" might not be a good enough standard -- allowing a team to train all three of their trainees in a throwaway scrimmage each week would make it far simpler than it should be. There should be some choices required in determining whether to train or not, and having a scrimmage game take care of an entire week's training just doesn't sit well with me. Then of course, a natural suggestion is that each game can be used for 48 minutes worth of training (plus any OT minutes), but outside of the guys who play 48 minutes in a game, that gets complicated.
I wonder if a better solution may be to allow any training type at any position, but perhaps with a bonus if the player is in the "right" position or a corresponding penalty if an extremely different one. Or maybe just a different allocation of the skills? For example, if you choose passing, for PG, there is a big increase in passing plus the regular, smaller secondary increases in the related skills. If you choose it for C, the passing will grow less, but maybe with bigger increases in handling and driving. It'd have to be balanced out so that a skill's primary position is always the best overall in terms of benefit, but that out of position training is close enough to be a valid choice. Of course, it'll never catch on, either with the "But I have to have 19-19-19 inside guys" crowd or the "no, a player without great potential trained perfectly is worthless!" crowd, but I can hope. ;)