I agree with nearly everything written so far.
You have to keep in mind that the Yankees had been the dominant team and organization in the game for 40 years by the time this policy started really hurting them.
While the Yankees had the added ace in the hole of running a de facto minor league team in Kansas City for some years, for the most part in the pre-draft era, signing amateur talent was the name of the game and they did it better than anyone, but that kind of success requires constant vigilance and the minute you start resting on your laurels and assuming things will always be the same, you're setting yourself up for a fall. They looked at the integration trend as a National League phenomenon, something of a cheap shortcut to success and didn't want any part of it.
The Yankee Way had always stood them in good stead and they were determined to stay that couse. When this finally started to show up as bad results in the on the field product (in the mid 60's when their replacement class of Pepitone, Tresh, Bouton and Linz couldn't step up to replace the aging Mantle, Maris and Ford generation), it was too late.
There is also the previously mentioned factor of them thinking their fanbase was upscale and white and their not wanting to draw black and brown fans. Bill's point about money is usually true but some people can, and this was especially true then, be truly shortsighted and obdurate when it comes to matters of race and class, even if it ultimately hurts their profits.
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"In my nine seasons, we definitely had some very high moments and some extremely disappointing moments. But one thing that I learned is this — once a Braves fan, always a Braves fan. No matter what. And as a player, that means more than you could understand." - Tim Hudson