The problem is that the MLB draft is so dreadfully dull. It's some combination of most (all) of the kids being no names to all but the most dedicated fans, and the commentators for the draft being MLBtv employees who won't say anything negative about anything. They don't even tend to have any of the actual players on hand so they can be interviewed. It was a struggle just to make it to the Storen draft pick before changing channels this year.
As you and most are well-aware I find it fascinating but to your point MLB does a terrible job packaging the draft. You are spot-on about the coverage on TV. There really is a need to get the top players together. There is no reason they cannot organize it around the College World Series and have the players participating in the regionals, Super Regionals, or CW to get together on site. They need to winnow down the number of faces/voices. To be honest, they do not need to trot out every single former major leaguer they have on the roster. Pick two and go with them along with two experts (Baseball America, PG Crosschecker, etc). And they need to get rid of the pollyanna from the scouting bureau. Every pick isn't great and they need to have a critical voice there.
The excuse is always people see these kids once and then they disappear for four years until re-emerging. MLB Network needs to leverage the forum they have to introduce these kids over time. Broadcast college games, the Cape Cod League, the HS all-star tournaments and build interest over time. They have 24 hours to program and they need to maximize it much better than they have.