Would you move??? I think I would. FSG are about making money not about winning trophies. Man City and Chelsea will do what it takes to buy the right players that will win them trophies. The success of players career is measured in trophies...That ain't going to happen until Liverpool find themselves owners similar to Chelsea, PSG and Man City.
No one can blame Sterling for wanting to move to win things. If he only becomes a bit part player at bigger club though, will he feel that has genuinely helped to win those trophies? Playing once or twice in a cup campaign must feel like you've been given a medal out of sympathy!
He's from the London area so maybe he's a bit homesick and would rather be nearer his family and friends. That also makes sense.