League of Legends

There are times when Sheen/Trinity Force are useful. And those aren't as many as when a normal AD build would be.

Unless you are flat out steamrolling the other team getting a TF early can lose you the game because the enemy AD carry will be able to output much more damage than you.

Why take Ezreal over other AD carries and not need to use Sheen/TF?
He has a cheap cost long range poke that means even if he's zoned out he can still poke and last hit to keep him in. He has a blink that can be used to chase, escape, goes through walls, and attacks after blink with very decent range on it. And before you mention Corki or Tristana, do note that Ezreal's blink has a shorter cooldown on it. I guess in tradeoff for its weaker range. Ignoring Trist's refresh, of course. Utility wise, it's much safer in general to chase with Ezreal's E than with Trist's/Corki's W because of it's range. If you chase with Corki or Tristana it is very easy to fly in too far. Whereas with Ezreal the distance is much shorter and the magic damage it throws out afterwards will cover the rest of the way.

Ezreal has a clear burst damage in lane. His kit just screams for him to move in and burst once the opponent is harassed to a point low enough to kill. This is easy to do also with his long range and cheap Q. No other AD carry has that except for Corki and he needs to get to level 6 to do it. Varus has an arguable one but it functions differently.

And you mention the AD steroids that other AD carries get but you forget that Ezreal has an AD debuff that essentially lets him win against a lot of AD carries. It can function like a mini Blood Boil for his teammates as well. In lane, if you go in to harass or for the kill and you land this on the enemy AD carry you're essentially ahead in damage now regardless of what they try to do so their only option in a lot of cases is to just back off.

He comes with a global ult. Ashe and Draven has this also but they don't have over the wall blinking and Ezreal can essentially have a constant 50% AS buff. They don't.

I can go on but I'll stop here. There is nothing wrong with playing Ezreal like a standard AD carry and if you think otherwise you're just talking out of your ass. I will even go so far as to say that if you rush TF either as a first or second main item you're doing it wrong (unless you're stomping the enemy team considerably). If you really want to take use of the proc then build a Sheen for laning if needed and then get a Bloodthirster afterwards. And to proc slow you don't need to get a Phage. If your team isn't giving you red buff when it becomes useful to have it then just work around it. And don't get Sheen -> Infinity Edge because it's much more difficult to cover the extra 800ish gold for IE than for Bloodthirster after dropping the gold for a Sheen already.
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