there are a few issues here that tend to get intertwined, but i think its important to look at them individually
sterling is 20. hazard came to the premier league when he was 21 i think. and it really took until his age 23/24 season (this year) to turn into the star many predicted. most 20 year olds are not superstars and a finished product.
hes english. so he is going to cost more than if he was italian or spanish. BUT, man united just paid £30m for depay, who is 10 months OLDER than sterling, and who has done all of his damage in a pretty poor league, relative to the big 5 leagues. so, given that united paid £30m for depay, who has no experience in the premier league and limited exposure internationally, what does that make sterling worth?
he has a few really important qualities: pace, passing ability, and good vision/movement. his finishing is meh, but again, how many wide forwards are clinical at his age in a top league? very few.
he can play 3 different positions effectively: CAM, LW and RW. and his idiot manager even played him as a wing back/wide midfielder. he is comfortable enough to play centrally behind the striker, and he is a good enough dribbler and passer to be an effective winger.
as others mentioned, the location could play a factor in his decision, and seeing as the arsenal team is loaded with his fellow english colleagues, im sure that wouldnt hurt.
i think its unfair to compare him to players like hazard, alexis, and others. he is much younger, he has much less experience, and hes played in a shitty team for most of his time in the first team at liverpool. last year he was great when he had suarez and sturridge firing on all cylinders. he'll have far more talent around him here.
if his ambition is just collecting winners medals, regardless of his role, then he'll probably put chelsea/man city ahead of us. if his goal is to win but also to play a lot of first team football, i think he'd do well to choose arsenal. he should just look at chelsea and what they did with cuadrado. pay a huge fee for him, then play him for 10 minutes every other game. and why would he want to go to the corpse that is man citeh? they look a team on the decline.
if we have the chance to sign reus or sterling, id take reus, because i think hes just as dynamic as sterling but his end product is more refined and hes in the peak of his career, like many of our current first team players. and he can play as a striker too. but, reus has had a bunch of injuries in the last year, and id be slightly worried about that. but i dont think signing sterling should be considered some huge step down. he clearly has a mountain of potential, and playing with players who are on his level or above, which he can do here, will help him elevate his game.
if we can get a better winger than him, fine, i'm on board, but who is realistically available that we can afford? its not like we're getting ronaldo, and of course bale is also probably a pipe dream if united go in for him. i mean id be happy if we spent our transfer funds on hummels, schneiderlin and a goalkeeper instead of another forward, but id be pretty happy with sterling here.