Unless some enterprising reporter were to dig around all the contract offers free agent starters turned down, us fans will never know. But it seems like common sense to a degree.
Yes there have been starters that signed, but they’re more like exceptions that prove the rule. Price was an overpay. Dice-K was a bidding process. Schilling, Porcello, Beckett, Sale and Eovaldi were essentially extensions originally acquired by trade. Arroyo, Wells, Colon, Byrd, Fister, Penny, Miller, Masterson and Hill were fliers. Dempster was 36 and in decline (and would have been a terrible contract had he not voided it). Julian Tavarez was a converted reliever. John Lackey was John Lackey.
Clement is somewhat of an outlier, but not really. Theo signed him after they whiffed on Pavano, Pedro signed with the Mets, and the Braves traded for Hudson. They also knew they were letting Lowe walk. Clement was reportedly interested after 2004, and said he wanted to throw to Varitek.
To take a non-Yankee example, the Dodgers were rather dumbly run for a lot of the 21st century, but in that same time period they signed Ryu, Hill, Maeda, Schmidt, McCarthy, Kazmir, Greinke, Capuano, Harang, Lilly, Wolf, Kuroda, and Lowe, and extended Penny, Billingsley, and Kershaw to multi-year deals. There are a few other factors there, but it seems like a significantly different free agent dynamic than the Sox have.