Alright lets talk about this a bit in depth.
Katarina has weaknesses due to her being a melee champion, yes. That is true. That is true for every single melee champion in the game. But using the Ranged vs Melee argument as a argument is invalid, because on that note. Ryze should beat Irelia. Syndra should be yasuo. These arguments are invalid when a champion is STRONG, or strong against other champions because of their Kit.
The only champions that truly counter Katarina are champions with MASSIVE AoE or Point Click CC.
LB vs Kata is a skill match up, a good Kata can avoid a LB Q RQ W. Or avoid Cass W. Avoid Swain Pull. These aren't counters, these are skill match ups.
Akali vs Kata, is actually Kata sided. Because Akali shroud doesn't do anything vs Kata Ulti. Every Single Match up you said was unfavorable is a skill match up. Minus Kass. Because he is a counter.
Now lets talk about Kata laning. Kata laning is comprised of three different attributes.
1: Waveclear and poke.
2: High Threat Range Extreme Mobility.
3: Roam.
Those three things make Kata extremely frustrating to play against, because even if you do out range her. She still has the means to get on top of you in an instant. You are effectively alway forced behind your caster minions, making wave clear for majority of mages very hard. Because you have to clear the casters after the melee, you are forced to stay in lane a bit longer. No single mage can freely walk up to the caster range and clear them without being harassed super hard by kata. Also going for the casters first put every mage into a position of free engage. If a syndra goes to use Q on the caster wave. At level 2+, that means that Kata is freely allowed to Q W, Pick up Dagger, W. Because now syndra only has one ability. Meaning that the trade will always be Kata favored. On top of the fact, that Kata never has to auto attack syndra. meaning that she doesn't ever pull minion aggro. If Syndra auto attacks Kata while within the range of casters, the casters will also deal damage to Syndra, which only exasperates the trade loss. That is only early laning.
A good kata will also be allowed to freely roam post 6. Because no mage is going to ever start by trying to clear the back wave, that is a 100% chance of losing 50% of your hp in one rotation to kata. Since majority of mages run Lost Chapter first item.
This is a big issue because free roaming at level 6 for the mid lane is a massive advantage to whatever team that mid laner is on. They get priority over whatever side the jungler and mid laner are at. If they decide to go bot side, even if your bot lane was winning a trade, it will result in a 2 for 2 at best. This is under the assumption that both sides are equally skilled. This plays into a massive snowball effect, which actually really helps kata do her job better. Because if Kata doesn't pick up those kills, she isn't a high prio for target selection for killing. because another player will be in a higher prio order. Teams with less than 3 CC will have a extremely hard time dealing with two damage threats. But that is hypothetical.
Now to somewhat help defend you.
Kata is a higher skill cap champion. 99% of players will make mistakes with her. Bad positioning, bad blinks, bad engages, bad everything. She is a very very high mistake prone champion, akin to Yasuo. So over nerfing her will result in a high diminish to the playerbase, similar to what happened to Ryze in high elo.
My thoughts. With the direction that Riot is currently going, I think that she will stay as she is, because I think it is within what they are expecting. But what we should see soon is a lot more grounding abilities. They made Poppy accessible to grounds, and made Cass ground better. I think we will see a period in which champions like Karma (W?), Morg (R), etc may have grounds tied to them.
If riot goes in that direction Kata will die as a champion, along side Yas, Kas, Irelia, Akali, Lucian, Ez, etc. We will see the tank meta come back even though they don't touch damage at all.
But hey what do I know, I'm just some random diamond player saying a random thought on a random post at a random hour, randomly.