Stan has stated that he knows the guys projected to go in the 1st round very well and watches them a fair bit. After that he lets Kelley and the team takeover.
He, and every other GM, know their top 40-50 ranked players very well. On average, they know the top 2 rounds of picks on a pretty high level. After that, it does start to drop off as far as how much they've actually seen, noted, scouted of the players themselves, and they defer much more heavily on their staff. But ultimately, their name is on every pick whether they do well or they flop. They're signing off on it.
I hate war-sports comps, but Ive used the comp before that a General isn't behind enemy lines himself scouting and spying on the enemy, their subordinates are, and they're taking the information of those people in the field to make the best decision going forward for the entire group based on that data and information gathered. It's their call, and the success or lack thereof falls squarely on their shoulders. An NHL GM is "in the trenches" a bit more than a 5-Star General, but they're largely examining ALL of the data and information received from their staffs and making the best decision, in their opinion, based on the data given and what they envision for the organization and its future.
So, yes, they do rely heavily on their staff, but it's up to them to have good people in place to make good decisions and gather good information, and to suggest they have ZERO say over any of that is just foolish, to put it mildly.
EDIT: It's worth noting, there isn't a GM in the league (hell, probably all of pro sports) that has 100% firsthand knowledge of their entire draftboard. They're pulled in so many directions in their day-to-day duties that it's not possible. They all rely on their staffs to do the lion's share of that work, which comes back to having the judgement and knowledge to hire good people for those positions. That, in and of itself, is a skill; hiring good people.