I believe that Centers slide in the draft for several reasons...
One is that there are only 32 starters in the league and only a few teams are actively looking for a pure Center at a time.
A second reason is that unlike a Tackle who needs his speed and full flexibility, most centers are not injury risks... being able to play through most minor nicks and scrapes as long as they maintain most of their power.
A third reason is that very few teams carry a pure Center as a backup and most of them use a G/C hybrid player who is primarily a Guard... again, cutting down on the number of actual centers in the league.
A fourth reason is that squat, powerful players tend to have lengthier careers than speed guys... again, cutting down on your need to find a Center very often. Look how long that Mike Pouncey has hung around. If he'd been a Tackle, he'd have been gone 2-3 years ago with his injury history.
So... in the long run, there are only so many openings in a year. Which makes this an easier position to fill, even though it is important. This is especially nice for us, because there are quite a few realistic Center options in this year's draft... meaning that some good prospects will fall.
I'm thinking that in a year like this one, it would be draft malfeasance for us to come away from the draft without taking one.