I think this depends a lot on the team and their needs amd the position of the player, though positions can be changed if you have the discipline and power at the plate.
One of my fav players, Hanley Rimerez, can't play his position well but man can he rake, drive in runs, etc. He just got a 14M contract this year and it wasn't because he fields his postion well, though he is young and will improve. I htink that his postion was changed to better suit the team's needs, which often happens (Arod for example).
If you are a power hitter or find a way to get on base, you get the job, if you can't produce runs than you have to be A+ at your position (SS or catcher for example). DH in college ball doesn't have to be a power hitter or one who fields their position well, often times just able to come up with the hit or HR when you need it, that is what makes them so special. Kind of like the Ortiz's of the world.
You practice fielding as a team and the team isn't always practicing. You are dependent upon that to improve your skills.
However, you can spend literally hours in the cages, getting better at what is perceived to be the most important thing for most coaches and managers. I have never heard anyone say, he can't hit for average but plays his position well. What I have heard is that he hits for average or above but needs work on his fielding skills. A lot also depends on the position you play. I know of one player who got drafted very early, not a power hitter, in fact not a HR in college(?) but way above average OBP, HBP and steals and fast as lightening as the CF. Excellent table setter as first at bat, every game.
STL's first pick was all about his bat. He may have been better suited for a AL team, but as stocky as he is, they say he is pretty fast on the bases. He can improve on his fielding, but his bat got him 2.3 million dollars. My son's friend was a 13 round pick, his bonus wasn't as high as someone who had 5 tools, but his bat got him that pick. The top draft choices, have the tools to get them that spot. Buster Posey, Matt Weiters for example, excellent catchers (with power arms) who can hit for power. There are not too many of them that exist.
Someone mentioned it and it's true, same for pitchers. You are more desireable if you pitch 90's,they can teach you to pitch. Same for hitters, you have that natural instinct at the plate and you are in, they can teach you your position, which sometimes is not even what you were chosen for in teh first place.
IMO, it's all about tools and you have to hit for one of them to get you noticed.
I'd like to hear PG's take on this one, he can shed a better opinion.
JMO