I did read what you said in bold, and I agree with all of your statements except for the part being designed as an assassin. At the time she was designed, there were no clear cut roles other than "DPS, Sustain/Tank, Support". This caused her kit to not be fitting as a mage, nor as an assassin though she's labled one due to her snowbally DPS.
Ahri is actually in the same kind of situation as Kat here as well.
The majority of people who say Kat is OP, mainly started after Leblanc, Kass, and Talon got their gamebreaking power removed.
Interestingly enough, like Kat, Ahri who is also considered unbalanced by many did a huge amount of damage more while being more sustainy than the current live version.
(Funny how they're both hated more after getting nerfed and losing a huge amount of burst damage without much of a compensation..)
But until the Trio of Midlane Op lost their game breakers which caused Galio to lose his small bit of popularity, Kat was a decent but not considered a OP/game changing pick. Because all three of them utterly brutalized these two into obscurity, once the silencing demons vanished people didn't know how to deal with Kat/Ahri because their big counters lost a tremendous amount of power.
And it spiraled downhill from there into the often seen "Kat is OP Rito nerf plz" or "Ahri is broken champ"