I'm a Kayle main and you are most certainly exaggerating. Kayle is no where near as broken as you're claiming.
Lemme see if I can help you by listing some of the things I hate when playing as Kayle.
Most frustrating thing, burst champions. Think Leblanc, Annie, Akali. I feel so helpless vs them because my ult is useless. Their burst often comes too quickly for me to even have time to ult and even if I'm expecting the burst, it's impossible to time it just right when they are about to burst you. I sometimes ult too early and they just wait it out and then burst me down.
Another thing. Kayle is hilariously bad in the early game. She loses trades to almost every champion in the game until she has Nashors. Before Nashors, she has no damage. I can get Nashors without getting a kill by around 12 to 13 minutes. That means before that, I'm not killing anyone without help and if I'm forced to 1v1, most times I'll die. This brings me to my next point.
She is one of the most item dependent champs I have ever played. Unlike popular champs like Ekko, Yasuo, Fiora, Darius, Jhin etc, who can start wrecking shit from level 3, she needs 3 complete items, usually
before she even starts becoming a serious threat. A Kayle with those three items is about as threatening as a Yasuo with
. Her heavy dependence on items to be strong is probably why she isn't as popular as Yasuo and others. The vast majority of the playerbase want to do nothing but fight fight fight constantly. No one has the patience to farm up while playing passive, except maybe Diamond players.
Lastly, Kayle is a terrible champ when playing on teams that feed a lot due to her item dependence. In such games, it's usually too late for her to do anything when she finally gets strong enough to make a difference.
So yea, Kayle is not broken.