The BOT obviously has power, the power the State of NC and the BOG has delegated to them. To deny that they have power is simply wrong on the face of the claim.
From a procedural perspective, Roberts could have clearly decided not to hire Belichick. But practically-speaking, as a Republican who has hitched his professional wagon to the Republican Party in NC for his current position, he would need an overwhelming reason to refrain from hiring someone with the CV of Bill Belichick because if he says no then every person from Preyer to Thom Tillis to Marco Rubio and all of their associates are going to wonder why UNC didn't hire Belichick for it's open position. And then Roberts is going to have to defend himself from whatever crap rolls back downhill as the repercussions of such a decision. So while Roberts had the technical ability to not hire Belichick, he would have been absolutely stupid to do so without an overwhelmingly good reason due what would have been the resulting fallout.
I'm sure that Belichick interviewed well, as he's a football-lifer and knows a lot about the game. I mean, it wasn't like he was going to walk into a meeting with Roberts and say, "Yeah, I'm one of the greatest coaches of all time but I'm at a point in my career where my 24 year old girlfriend leads me around by my dick and I plan to load my staff with my family, my underachieving buddies and their family, and whomever else I know sitting around the KFC while firing everyone I don't know without good reason and I'm so arrogant I won't recognize that I'm in entirely over my head until my team gets its head crushed in on the field numerous times."
So, in short, by the time Preyer got involved and offered Belichick the job and ran off all other serious applicants, Roberts had very little real world choices other than to hire Belichick if he valued his own position.