Katarina has a poor laning phase but doesn't really fall off late, when played around her in solo queue at all ELOs she can carry games for teams very easily. She can roam bot lane and very easily pick up kills and during the later stages of the game in team fights, if your team focus's out the enemies with CC and keeps them CC'd or away from Katarina she can deal huge amounts of damage very quickly. It seems like in an environment like the LCS teams would have the synergy necessary to keep champions that can shutdown Katarina in fights off of her, so why is it that she is considered a poor pick and see's very little play? Why don't teams try building a CC heavy composition around her similar to what was done with Kog Maw a lot recently?
The main strength and problem with Katarina is her ultimate ability. Her Q, W and E abilities have very low AP scaling. For Q it is 40% scaling for initial cast and 20% scaling damage bonus when the mark is consumed. For W and E it 25% with 60% AD scaling on her W. Her ultimate also has a 25% AP scaling for every dagger. She throws out a dagger ever 0.25 seconds for 2.5 seconds.
2.5 / 0.25 = 10
This means that if someone stands in her ult for the full duration she will inflict 250% of her AP as damage on to the enemy.
If Katarina gets CC'd instantly, no damage. If the enemy builds a team composition to counter her then you are left with a champion that has a bad early game and horrible scaling. Pros will be able to do this effectively meaning that choosing Katarina results in the champ being useless for the entire game.
You brought up Kog'Maw which is a valid point. The difference between Kog'Maw and Katarina is that Kog is ranged and deals mixed damage. Both of these greatly increase his survivability and damage during crucial late game teamfights. Kog being ranged makes it a lot easier for his team to protect him. On the other hand, Katarina has to dive into an entire team to get her damage out. When Kat dives, the enemy team can easily CC her and instantly burst her down. Also Kog deals mixed damage, both Physical and Magic Damage making him great at dealing with tanks. His percentage health damage also helps with that. Katarina does mainly Mainly Magic Damage making her easier to itemize against her and stop her in teamfights.
To wrap this up, Katarina is too one dimensional and her low AP scaling and need for resets punishes her ability to be played in the LCS. she is also outshined by hypercarries like Kog'Maw who do much more damage and are able to survive for longer.
You've actually listed precisely the reason why she isn't used in professional play. Yes, Katarina can snowball like nobody's business but hard CC and coordinated teamplay, which professional teams will have in loads, can and will shut her down. While Kog'Maw faces the same issues, he has the benefit of range whereas Katarina needs to jump into the enemy team to deal her massive ult damage.
While I agree with everything @Yuuki has said, I think it is important to point out that it takes one knock up, stun, silence, etc. to make Katarina almost useless in a fight. If her ult gets interrupted, her damage contribution drops very significantly, and she needs the damage to get resets. For her to get a full channel off on her ult, the other team will have to be cc'ed for a lot of the time, something which can be very hard to do. Basically, she is a bit too one dimensional and easily countered.