It's not really mainly about the Athletic testing from Pro Days or getting a 40 time, it's about getting tangible Medical clearance on players that still have concerns with Surgeries during the off-season... and also about players with not football related issues, they need in person sit downs to get a better picture on the mental aspect of a prospect.
Some teams have extremely strict policies regarding Medical, and Psychological clearances on prospects.
Some teams will not even have Tua on their Draft board if they don't get the physical MRI slides of his Hip joint looking healthy.
A guy like Linebacker, Willie Gay Jr. is another example.. he has 2nd round speed and talent.. but he punched a teammate in the face, and cheated on a Chemistry test... a GM needs more background information on what happened to prompt those decisions and if this is just an example of a kid making a bad mistake, or if he has legitimate character issues that will follow him into the league before they will even consider drafting him.
These is vital information that directly effects the decision making of building your team for the future and it cannot be obtained right now, and GMs have a legit case for getting the Draft pushed back a few weeks if it means they can obtain this info.