Fighting outside zones of power doesn't really work all that great when Kirylis's zone of power is the entire lane.
Trundle's zone of power is pretty nasty, but he still he to walk within it. He doesn't get the ability to freely dash around and prevent escape or ensure his own easy escape. Not just that, but generally Trundle would have to engage his foes in combat to get something done, or at least have buddies around.
Not so much with Kirylis, who's able to lay down her W, dash over a jungle wall, press E to pull someone through a wall or otherwise out of the fight, then dash back into the fight with the enemy team one champion short - or just stay there and beat the poor sap (probably an ADC) to death. Cooldowns also become less and less a balancing point given Kirylis' preference for autoattacks and AD letting her splash into an Essence Reaver-centric build that would let her fire off her Q at alarming rates while also getting her zones of power back online even faster. Specifically, at 30% CDR, you've got a downtime on the Q of about 2.2 seconds. If you can squeeze in an extra 10% from Ionian Boots, you've got 1.6 seconds of downtime. Maybe even 1.3 if you're daring and skip out on the Ferocity tree for the CDR.
The slower speed on the E doesn't quite work out as a balancing factor since it starts at max range. I like the kind of paradigm there, and in fact Kirylis was one of three champions that used that in this contest, but with a (potentially AoE) pull to your cast point, that's insanely powerful since a max-range cast can't miss.
There's many things crazy about the ult. The damage alone is pretty overwhelming - that is a TON of damage you're doing over the duration, more than most ADC ults. The slow alone is pretty powerful too. The dash cancelling is potentially gamebreaking. These are all powerful effects already, and now they're on an effect whose range is so huge you're trapped inside it because it ate the lane. You can barely Flash out if you're trapped in the center. How can anyone teamfight Kirylis's team if this ult is up? You can avoid Amumu's ult since he's pretty obvious about what he's trying to do, but Kirylis can just dump this at any time - and almost anywhere because of her W - and trap an enemy team.
The statement that Kirylis needs someone to initiate for her is...not entirely true. Maybe if she's saving her E, but she's perfectly capable of initiating a fight by herself.
Probably the biggest shortcoming here is that, even though you planned an assassin, I can definitely see a support player taking Kirylis for the huge amount of CC she brings to a team. That's...not something that should be happening on an assassin.
If you had to do it again, I'd recommend you erase the E and R and the on-hit damage on the passive. Probably the fresh snow mechanic too - meaning the heal on the Q goes as well. You'd want to shrink the W's range, since it's a workable ability but not with that huge an area. For replacements for the E and R, it seems like you'd lean closer to isolating a single target and dismantling them while they're stuck with you. The basic idea of the E (pull people closer to you) could probably work with much less range, since Kerrigan does it in HotS, but you could also go with creating a directional wall of spikes facing toward you that stops and damages anyone trying to dash through. For the ult, if you're attached to the snowstorm kind of theme, you could mark a target champion, and after a delay unleash a gale that damages and shoves everyone away from that champion, thereby leaving you with a fresh target.
You'd end up with increased team utility over other assassins to make up for your lack of upfront burst, but the power here would be distributed in a more selfish way that incidentally helps your team instead of universally-useful utility.
As always, though, I'm looking forward to your next concept.
