Exactly, it was a good move for him because he found a manager who valued his specific skillset and he has been part of a successful era at Liverpool. If he had stayed I find it very unlikely he would have suited a Pep side with his preference for more technically gifted midfielders. Sadly peoples view of Liverpool is skewing their view of Milner.
The narrative that he left for playing time (& they specified the midfield slot) doesn’t stack up when you look at the stats. It was a narrative cooked up by LFC, Milner and the media. It was another way to have a pop at City - old school players take a pay cut to leave the club for the glory of the game. It’s that moral high ground the cult on the Kop love to take.
The reality is he wanted a long term wedge, good on him for getting that security, I don’t see him as a greedy bastard, but I don’t like the bullshit. I don’t boo players, but am taking a guess that those who do boo him saw through the story that he moved for his need to call central midfield home.