No, I don't think it was safe to trade down and pick Mayo. BTW, I have not criticized either strategy.
It is NOT always safe to trade down. Specifically, with regard to Mayo, Belichick took a very serious risk when his player was there at 7. He traded down, secured an extra pick and paid the player less. He won his bet. He wouldn't have been so celebrative if someone else traded up and drafted mayo at 8 or 9. We had a critical need for an ILB. Belichick took the risk and won.
In 2009, Belichick had at least one player he wanted who he thought would be available at 23. He had the 34, 47 and 58 for his other needs. Belichick had decided that we badly needed a #3 receiver, and one who perhaps would get more and more reps, especially when either Moss or Welker eventuially left. He chose NOT to move a couple of spots to make sure of getting his player. Harvin was picked by someone else at 22, and we are still waiting for our #3 WR, although Belichick did end up using a 3rd and a 7th and sign two free agents to try to meet that need. So this time lost his bet. He ended up trading down and used his extra picks for the two wideouts he did end up drafting.