I think you underestimate the clout that the Red Sox organization has. The Sox are much closer to the Yankees stature than the Royals or Marlins are to the Red Sox. I didn't say that the technique would work for everyone. In fact, my post addressed only the Yankees use of the technique, not the Red Sox and certainly not the Royals or Marlins. I discussed how the Yankees have been effectively employing this technique to keep the cost from spiraling too wildly upward.
I do think that the Red Sox could effectively use this technique at times. You mention Teixeira as an example, but apparently from the information out there, the Red Sox were willing to go higher, but walked away before giving the final offer. What they did really was really confounding to me. (I realize that a lot of what is in the press has been distorted along the way, so it's hard to know the true facts). The Sox made an offer. Tex countered. Sox walked away and declared negotiations over in the press. Yet, in stories that came out later, they were willing to increase their offer. If you are going to employ the tactic of making the best and final offer that has to be accepted on the spot, then make the final offer. Don't expect that you will be able to get back to the table with the new offer.
Competing with the Yankees for players is challenging, and overall, the FO does a good job. Good negotiation skills and techniques can help close the gap. I think at times they are penny wise and pound foolish. They'll miss out on a good player like Abreu because they wanted the Phillies to eat $2 or 3 million more than the Phillies wanted to, but then they throw $39 million at a useless dog like Lugo and they misjudge the market for Dice K by paying more than 25% more than any other team was willing to bid. The FO is a smart bunch. I think they are good at assessing and learning from there mistakes. While some of us give them a free pass when they screw up, I think they are quite a bit more self critical.