Lulu is a good counter to assassins, but so are Vi (Ult to lock them down), Zilean (Ult to revive), Graves (close range burst and durability, especially with a support protecting him), Alistar (strong peel), Janna (dat disengage), Lissandra (ult provides both ends of the spectrum to counter assassins), Maokai (peel and damage reduction), and Shen (strong shield and peel). Of these, Lulu, Vi, Graves, Janna, Lissandra, and Maokai see consistent play at the competitive level.
The reason Lulu doesn't crowd out these champions for positions is partly her rather low damage without itemizing for it and her natural squishiness when she's not itemizing for it. You can turn her into a glass cannon, deal good damage, and use your support abilities to try to keep yourself alive, but if your ult is on cooldown, your ability to protect yourself drops drastically. Or you could turn her into a supportive bruiser, deal decent damage, and be an annoying lane bully who provides support later into the game. She cannot tank or engage like a Maokai. She cannot reset a fight like a Janna. Her damage is going to be lacking when compared to Graves or Lissandra. And her jungle is much, much weaker than Vi's.
Basically, the reason you don't see a ton of her is that she has drawbacks, especially when compared to many of the above champions. You're overestimating her strengths and underestimating her weaknesses. She still sees play because she's still a strong champion, but she's not the top of the heap.