I main supports. I do a job good enough, by feeding my AD carry, denying the other bot lane, roaming or pulling off clutch support moves, to "carry" the game more often than not. The one exception to that rule is Ezreal. Unlike other AD carries, I don't feel I make anything happen with Ezreal other than hope he is good enough to out-play our opponents.
From my point of view, he has poor harass and poor burst. I can't play a kill lane Leona or Alistar as we'll likely get out-DPSed. I can't play a passive Soraka, but it feels like we're getting denied too easily. Blitzcank is about the only support I have had some success with, but even there it feels considerably more difficult than when I play Blitzcrank with other lane partners. Ezreal really feels like laning with a bad Corki.
Best Answer
The most important part of support is going to be the communication. This needs to start before you select your support. You won't be able to pick the right support if you don't know Ez's playstyle.
Some Ez's might want to go aggressive. You're right about Ez lacking the ability to blow an AD up quickly. However, his poke is significant. Pair him with a Sona, especially if you ban out Soraka, and you can force the enemy out of lane. Many Ez's though prefer to sit back and farm, doing damage once he's completed a solid part of his first damage item. In this case, Soraka or Jana will be a better compliment.
Recognize that the game isn't going to be won in the lane. It's going to be that you enabled him to farm well enough to be a force in team fights. Just because you're not getting kills or doing anything exciting doesn't mean you're a bad support.
I also prefer to get exhaust over heal or CV. Lots of people want to dive in on Ez. Having an exhaust to help peel for him can be crucial.