I think the whole "can't handle City X" thing gets thrown around too often. And it becomes the reason for every bad season.
For example, slasher and I argued that Travis Shaw clearly could not handle the "pressure of Boston," despite his being a relative no name player himself much like many of the names you mentioned. While he did better in Milwaukee, was it really because of the reduced pressure, or because that tiny bandbox of a ballpark sees a lot of home runs that were really just routine flyouts at Fenway? Yelich also did much better there, but it wasn't because he escaped the high pressure Miami team. Despite what many Sox fans think, Milwaukee is not some little podunk town where professional athletes mingle with the normies at a random corner tavern, and the press is far more concerned with local zoning ordinances than with the Brewers, Bucks, and Packers.
And many of the Yankee players I mention - Hicks and Grigorius being two prime examples - were actually fairly highly touted prospects who also struggled in some markets we would consider to be "low pressure." But they excelled in NY. In both of those cases, the ballpark itself was probably a bigger factor, as it worked with their left-handed power strokes (although Hicks, being a switch hitter, didn't get to exploit it full time).
I think most of the Yankee success stories were also helped by the deeper lineups. You can't pitch around everyone, so once pitchers have avoided Judge/Stanton/Sanche,. they might not have even strategized on how to handle the Urshela's and Tauchmann's s of the world, and those hitters feast on lesser or a complete lack of preparedness.
There are far too many factors in why a player struggles or succeeds in one city versus another, and pressure is probably rarely a factor, or at least one that occurs less often. In fact, Cole arguably faces less pressure. Worst case scenario - he goes down as the biggest free agent bust in professional sports history who still makes nearly a third of one billion dollars. That's a pretty soft cushion to land on when you fail.
If Cole has a downfall, it is going to be his over-reliance on a very hard slider taking a toll on his elbow (no pitcher in MLB throws a slider as hard as he does with the same or greater frequency). And even then, he still makes $324 million dollars. He's clearly a strong-willed competitir who wants more than just the money - but they all are. And the truly great players treat failure as a motivator, and that attitude is a big factor in even reaching the majors. The minor leagues are loaded with amazingly talented players who get in their own way and therefore barely ever crack the ranks of being one of the best 750 players in the world.
If people are couting on Cole fading under the bright lights- something he didn't do in the postseason, you're very likel;y to be disappointed. If you think after 1 or 2 seasons, tthat he is going to start seeing the IL a lot more often and see his velocity drop into very hittable numbers, you have a much better chance at being right about him.
I believe that players actually face the most pressure in the minors. Because if you fail there, you stay there. Think about the best players you played with in high school or college. How many made MLB? And then how many saw their careers just end in some obscure minor league city you might not have even known had a team?