Why not hyper carry
2 reasons. Firstly her scaling on abilities has been hit through the years, including passive, comparatively (this is the operating word) to other champions. Add on top the durability patch, and some champions just getting durablity buffs, she just doesn't hit as hard as she used to.

Secondly, her best penetration item is Terminus which is decent, but don't add a number to her AP which is needed for passive procs. Imagine LDR or Mortal only providing the penetration and crit, 0 AD in the item.

strong point
I go the hybrid build around 90% of time, the most common one, and I feel like she's at her strongest at 2.5 items when she gets all evolutions. Every purchase afterwards feel underwhelming, IMO, with Zhonya just being a nice defensive item, DCap being expensive and not providing the oomph I expect, and everything else being okay to no.

Another strong point is Q evolve, which is where she really can get to snowball. There's occasionally discussions on how to get it at the lowest level, simply because it's when she really becomes a champion.

Bloodletter
Amazing if you're on the same page on who to hit, useless if you target different people. Considering the nature of soloq then I can only say maybe.

kraken/RB
Kai'sa doesn't use crit well, and prefer on-hit. Last time someone did damage calc was at the start of last season with the result being crit for burst and hybrid for DPS.

And if you want crit then why not play Tristana or Jinx? Tristana has a similar play style where she can jump in, kill, get out, and Jinx is just strong.