Imaro
Legend
I'd also point out that there are at least three reasons why Imaro's example of a fighter doing AoE attacks doesn't really work.
I disagree... my example of the fighter doing AoE attacks works fine, it's not breaking any rules and is possible. That said I'll address your specific points...
1. In order for the fighter to attack 4 different targets they'd either have to all be within 5 feet of the fighter or he is one shotting each one. Otherwise, he's taking an opportunity attack from each opponent other than the last, as he bounces around the battlefield. AoE attacks typically don't open you up to multiple opportunity attacks unless the caster is really desperate.
Well the Fighter has the highest AC and highest hit points plus Second Wind to mitigate the attacks and damage he might suffer... but I would assume he would use a reach weapon if he's going for AoE... so now there is no need for the fighter to move into the threat range of most creatures, but instead move along the outskirts attacking but not taking attacks... Let's also take Polearm master which gives us another attack for d4 every time we take the attack action and attack with our glaive but also allows us to attack any creature who enters our reach... Oh and if we want a wider area of effect take the Mobile feat I commented earlier tactics and strategy were necessary to pull this off but I didn't think I'd have to construct a step by step...
2. If the fighter is not one shotting each opponent, he can choose to double up attacks and keep attacking the same target in order to put it down. AoE effects cannot do this. You can't choose to fireball three ogres and then triple the damage to one ogre. It doesn't work that way. IOW, not an AoE attack.
Wait what... your complaint is that in 5e he's versatile enough that mid-attack the fighter can switch from doing an AoE attack to focused damage if he chooses... He's supposed to be a master of combat why is that a bad thing again? I don't even know what to say here... that's been our point all along, the versatility and flexibility inherent in 5e... I don't have power that forces me to act in an exact way with an exact outcome... instead it's based on how I choose to use it in play and in the moment...
3. The fighter can do this once per short rest. Meaning, at best, he can do this once per encounter, although, in play it's likely once per two or three encounters. I'd say that's a pretty poor controller that can only make one single AoE attack every two or three encounters. Heck, one that could only do it once every two or three ROUNDS isn't a very good controller.
I'm not going to spend the time constructing another build to prove this wrong but I'll give you a hint you start with the Battlemaster and pick the right weapons and feats... enough said.
So, no, fighters aren't controllers in 5e. They can't do it.
No they aren't but not because they can't do AoE attacks... it's because they are versatile enough to move beyond a "controller" role.