Interesting points about extra/external motivations for Roger to keep going. Though by all accounts, he really does enjoy the tour. And Wimbledon hasn't even been played since he nearly won it. (Excellent point that he would have beaten Nadal and Djokovic b2b in that one. Hadn't thought of that bonus.) My pop psychology take on the 3 and their attitude about their GOAT race is this: I think Roger spent the majority of his career not even expecting to be in one. However, he did tip his hand about caring about records when he actively went after #1 ranking again. I also think that winning that Wimbledon meant a lot more to him because of the competition for top records. Did he really need a 9th Wimbledon? Of course he would want it, and he's competitive. But I always felt that the fumble at the goal line was due to the extra pressure of Nadal having 12 (at the time) RG titles, and to knowing that he probably needed one more to secure the top spot in the Majors race. I wasn't sure if I thought he'd play, say, small tournaments to pass Connors for most titles, but now that you suggest reasons he would stick around, then why not? Plus, he's always seemed to like going to places he's never been to play. Now I can see why that might also make Uniqlo happy. Suddenly I can see why that might make sense.
My take on Nadal is a little different that @rafanoy1992's. I believe he means what he says. He says that he'd love to finish with the most Majors, but he won't lose sleep over it (implying that it doesn't matter as much to him.) Both things can be true. Having the most Majors (of all and of all 3,) is a form of winning, and they all really, really like to win. But it has always worked for Nadal to keep his eyes on the immediate goals, and I don't think this is false modesty or a laissez faire attitude about the GOAT race. Just a different approach, and I do honestly think that he's not going to change the way he approaches the work at this stage in his career.
Novak has flat-out said that he wants the records. Even Rafa addressed that. He said there's nothing wrong with it, it's just a different approach. I agree with that. It's not more venal to say it out loud. Whatever motivates you to greatness and to keep producing it, good for you. Maybe that's why he seems more testy of late. He's got a lofty goal and he's the one playing from behind. Bottom line, whatever small variations in the ways they think about the GOAT race, I'm each of them would like to be the one sitting in the last chair when the music stops.