1. The Catalyst is using synthetics to kill organics...but this is the problem it's trying to solve!
I never subscribed to the "Yo dawg" meme. It's true that the Reapers destroy civilizations to preserve organic life as a whole, so there's no contradiction there. It's a subtle difference that even some of the writers seem to forget at times, considering that some characters (most notably Shepard) sometimes slip into claiming the Reapers will destroy all life.
So yeah, it's methods follow it's stated goal. The problem though is that the stated goal is ridiculous. There's no mention, for example, of why the ideal of organic life is worth so much pain and suffering. Causing a cycle of unending genocide to avoid a single instance of genocide is not logical.
In other words, is it better to have a race of machines rise up and eliminate all organic life once, or to let organic life grow into civilization, and systematically eliminate these time and again? The reaper's solution actually causes far more suffering and death, by allowing life to recover between cycles.
2. In my playthrough, Joker/EDI hooked up and the Geth/Quarians found peace, therefore conflict isn't always the result! Several arguments can be made against this. First, giving two examples doesn't talk about the bigger, overall galactic picture (winning a battle doesn't mean the war is won, so to speak). Second, we haven't reached that technological singularity point yet by which creations outgrow organics - basically, when synthetics will normally come to dominate the galaxy. Third, evidence for the synthetic/organic conflict is there in the past - in the Protheans' cycle (Javik dialogue) and even in previous cycles (the Thessia VI says that the same conflicts always happen in each cycle).
Let me point out all the errors in your reasoning:
First, the Catalyst is the one making the claim that 'the created will always rebel against the creators'. Taken as a logical construct, finding a single case of contradiction proves it to be wrong. If I claim all cars are blue, and someone points out one car is red, I can't say I'm still right because maybe a later owner will paint it blue. EDI and the Geth are not proof that no synthetic race will ever rise up and destroy all organics, but they are proof that the Catalyst is wrong.
Second, even the proof you mention backs up the claim that the Catalyst is wrong, since the conflict appeared before and every time the organics prevailed. And in no example were the synthetics shown to be motivated by the desire to destroy organic life.
Third, you can't say that my circumstantial evidence doesn't count but that your circumstantial evidence (which I showed above to not support your claims anyway) is absolute proof. Logic doesn't work that way.
Fourth, the Catalyst has no proof that his claim is accurate. For him to have any proof he'd need an example of a single instance where synthetic AI grew to hate all organic life, overpowered it's masters and destroyed all organic life. Since this has never happened (else there would be no organic life left), the Catalyst has no proof. All it has is an unsubstantiated, easily proven false assumption, which it has used to wage a campaign of genocide on countless civilizations for millions of years.
3. If synthetics are the problem and the Catalyst is trying to protect organics, it should just kill Synthetics instead!
I'll concede this. No one wants to have to obey the rules of some alien monster that thinks it knows better than you, and if allowed to continue to develop the races will eventually surpass the technological advancement of the Reapers.
You mention that the Reapers value diversity though and I disagree there. That's never stated or hinted. Apparently they're perfectly happy to create a human reaper and let the batarians, asari, quarians, turians, krogan, etc go extinct. Not to mention all the species of non-sapient lifeforms that go extinct when they destroy the biospheres of the planets they're reaping.
Even the idea that they're trying to encourage cultural diversity falls flat. Sovereign very clearly states that they left the technology behind to encourage the races to grow along very specific lines, so they could control the cycles more easily. That's going to lead to the same patterns repeating over and over, countless races retreading the steps of those that came before and finding the same dead end. The Reapers actively stifle innovation, by their own admission.
4. The Catalyst should've done Synthesis instead of Reaping in the first place! First, doing synthesis may stop new life from flourishing by the Reapers' logic (see leaked script above); without clearing out more advanced races, younger ones might not be able to develop freely. Second, the Catalyst would've needed the Crucible. A pseudo-argument (i.e. not based on fact from the story, but interesting) can be made that the Synthesis was the long-term solution but the Catalyst would only enact it when the galaxy was "ready" for it by building the Crucible.
As shown above, the Reapers don't care about races developing freely. This is not their intention, and they actively work to avoid it. A race that developed spaceflight that didn't depend on the relay network would not be fragmented when they take the Citadel, after all. So no, this point is not something you can cling on.
"The Galaxy being ready" is a ridiculous claim. Everything in ME comes down to a single exceptional individual getting impossibly lucky. That's not a measure of galactic readiness. Or if it is, it's a very poor one. If you're waiting for the 'right time' to do something, it shouldn't be based on wether one person can accomplish the impossible. "Oh, Shepard tripped and broke his neck. I guess I'll have to kill everyone and try again. Pity, these guys looked like they were so close too." Not to mention, the Reapers always give the races of the galaxy a set amount of time, they're not looking for signs that they need to be either reaped or ascended. Although the most obvious way to show that reasoning to be false is that as soon as Shepard gets the Crucible in place, everyone is affected. All the developing races that would have formed the next cycle included. If you can ascend races that obviously aren't ready like that, why wait for anyone to be ready?
To be honest, the whole Crucible is another ridiculous contrivance. It's better not to try to think too hard about it.
5. But...the Catalyst is justifying genocide! It doesn't view it as genocide. Rather than exterminating species, it believes it's preserving them and even stopping them from being exterminated or enslaving/exterminating others; arguably, it believes it's doing the exact opposite. But of course, it is actually genocide, and we should try to stop it. Just because the idea of what the Catalyst is doing is evil doesn't mean that its logic is flawed. I personally don't agree with its methods, but its reasoning seems sound.
The Catalyst would have to be very deluded to think it's not comitting genocide. It is perfectly aware (or would have to be, after millions of years of doing this) that the percentage of people that get reaped is a very small one, and most people die fighting the reaping. The people turned to husks and sent to kill the living are not getting reaped, for example.
We're not told how the reaping works, in fact, so it may well be aware that even the ones that are reaped are killed in the process, and nothing of the living person is left in the reaper. What the Catalyst definitely thinks is that his genocide is justifiable. I don't see how this is a point you felt the need to argue though. People think the Catalyst is a monster, and you conclude that yes, he is a monster. Explaining that it doesn't think of itself as a monster is beside the point, and does nothing to justify its actions.
6. Wait, Sovereign/RannochReaper told us we couldn't comprehend them, but I understand this!
There are two ways to interpret what they said. One is that we actually couldn't academically comprehend it, in which case they must've been lying or it's just bad writing. Another is that we couldn't possibly comprehend the magnitude/scope of it, which is true. A human with a lifespan of 150 years (canon) can't comprehend hundreds of millions of years of organic evolution and stuff.
This is a literary device gone wrong, not something to argue about. The writers wanted to have their cake and eat it too. A lovecraftian horror from outer space that we can empathise with. If the motivations of the Reapers are too alien to be understood by humans, then the writers could not understand them either. If the writers can understand the motivations, then any human can. The way to get away with this is to never explain the monster's motivation. Better yet, never have the monster acknowlodge the human characters at all.
No human writer can write a character that is impossible to understand by human minds. Bad writers think they're so far above their audience that they can pull it off, which just makes their awesome monsters fall flat.
I've only focused on the points you brought up, but there are plenty of other issues with the Catalyst. One could ask why it feels the need to create Reapers, for example. Who is it preserving these civilizations for? A museum that no one can enter has no purpose. But that's probably more thought than the writers gave to the issue.