I know we’re all Kentucky fans here, and thus can’t stand Duke, but I’m on the side of not finding Battier’s story at all unreasonable.
Take sports fandom out of it and imagine you have a couple different employers recruiting you for similar positions. You express a preference to discuss it with them at a particular time. One calls you during that time and gives you their undivided attention. The other calls at a random time and is clearly in the middle of doing something else. I have a hard time believing that the first one wouldn’t make a much better impression on anyone. It may not have actually been the deciding factor in his recruitment, but if anyone was being a diva in the story as told, it was Pitino, not Battier.