Current thoughts on poppy, mind giving me your input?

I have played her several times now and seen her played just as many, and I have to say... Something about her feels very bad and unsatisfying.

So much of her power seems to be loaded directly into the utility on her ult, and when her ult is on cooldown she only feels like a partial champion, she feels as if her entire role is just non-existent. Not only does her ult feel like it has a very long cooldown, the actual situations she can use it in feel lacking since it's a very small AOE, and it will always knock an enemy back towards their base. I feel like her ultimate either needs to be off cooldown more often so she can take advantage of those strategic moments when she can use it without fucking over her team, or else the rest of her kit needs a bit of a buff so she doesn't feel completely gimped whenever it's not up.

She feels too niche, her actual moments of satisfying power are spread out too much, as are the times at which she feels useful and impactful compared to other champions.