Sorry, I don't think I said that the Patriots let their players walk, but if I did I retract that statement! My belief is simply that the Pats don't seem to sign even their most valuable players to contract extensions when the players have two or more years left of their existing contracts... they tend to let the players play out the contract, at the risk of having to eventually pay a premium and face difficult contract negotiations, like they did with all the players you listed except for Warren. Yes, they did resign all of these guys, and I want to say that they extended Warren before he approached his last year, which I though was brilliant. Same with Koppen and Light, Wright, previously Faulk, possibly Sanders, and I'm sure there are other exceptions. But the negotiations with players like Mankins, Seymour, Branch, Samuel, were just ugly, and even Brady and Wilfork weren't happy during recent negotiations in the last year of their contracts.
This approach is validated in players like Law, Moss, TBC, etc, but I think there are plenty of instances where you can say... gee, if they'd just extended his contract before the final year (Brady, Mankins, Wilfork are the most recent examples), they would have payed less over the long term of the contract, at the expense of higher up front costs. Obviously they can't do this with all of their players, but sometimes it does seem contrary to their long term team building approach.
With guys like McCourty and Mayo, you just see someone like the Raiders/Asomugha and Steelers/Timmons setting the market first, and then the Pats having to face paying a premium that they don't want to pay. I applaud them for get a new deal done with Brady *before* Manning, but I shake my head that they let the market get set by Evans and Mangold and getting in the situation that they are in with Mankins.