I agree completely with what you're saying about him being in his prime and how it's the only concrete move for him right now, but I think you are being overly pessimistic about his chances by saying the move is 'the last one available to him'. Yes, Madrid are unlikely to move for him, and many other European clubs are also set, but he has just had possibly the best season of his career at a struggling club.
Despite his 'baggage', he is still one of the best players in the Premier League, and I'm sure some other club will move for him this summer, or in January. You have to remember, some clubs may spend for the sake of spending, or even for intent, and signing Suárez would really signal intent. Chelsea, Bayern and the two big French clubs could possibly be interested in him, should Liverpool relent in their efforts to block his move.
Chelsea need a proven goalscorer at this level; Bayern have just lost Gomez, Lewandowski is not joining this summer and they are known to splash out (sometimes needlessly); and PSG and Monaco may want to further bolster their strikeforce. You talk about them having their pick of the world's young players, but who says they want to buy young? Generally, tycoon-owned clubs want immediate success à la City, so they splash out on established stars.
If you think this is tl;dr, then here's a summary: it's naïve to think clubs only buy when they need to, and he is not exactly in the worst form of his career, despite his misgivings off the pitch.