Her outplay potential is the reason for her problems. Nerf her attack speed too much and you destroy the champion. She just has a huge performance spectrum. She can do everything. She has engage (kind of with her ult and knock-up) she has unlimited lane poke, (if you get 2 kills with the E for the 100% mana refund) she has her passive for insane kiting, and has potentially very high damage. She might lose to a Lucien in burst 1-2sec, but thats it.
If I were part of Riots balance team, I would leave Kalista completely alone and instead I would do 2 things. First just like how Zed gets an icon when he has dealt enough damage for the killing blow from his ult, so should Kalisa have a similar icon when a champion has enough spears to deal the "killing blow".
2nd I would look to nerf her laning phase. It is potentially too strong when you look at her team fight potential as well. 1st. I would completely remove the mana refund from her E unless it only deals damage to minions. She is the only ADC in the game that I know of who has the potential to deal "free poke" with no mana cost. Next I would tone down her Q like they are. If Riot really wanted to do one more thing, It would be her ult. When she picks someone up and that person picks a location to go in, there should be a 1-2sec delay and a splash graphic to show the target area.