Kaworu wasn't lying. Humanity blames Shinji for the current state the World. In 3.0+1.0, Midori says this to Shinji:
Midori: "You're a disease! Do you know what humanity's been through thanks to you and your Near Third Impact!? This is all your fault, you and your father...! I won't let you do it!"
I's true that Shinji is blamed for the N3I, but the N3I seems to have a connection with the current state of the World (we just don't know what is the connection). So, Shinji is blamed for the state of the World as Kaworu said.
Here Shinji is once again blamed for Near Third Impact specifically, as it's the case in every single instance where he is blamed for the destruction by anyone except Kaworu. Even at the Wunder where Misato explained Shinji's responsibility by even showing him footage of the event, she didn't even bring up actual Third Impact, which makes zero sense if we was also responsible for it due to its much greater magnitude.
Now, going by the translation of this specific quote, Midori is also including Gendo here, so it makes sense why she talks about how much humanity in general has suffered.
Another proof that Kaworu wasn't lying is because he's the only character to explain to the audience what is the N3I. Other characters don't explain what is the N3I, they only mention that N3I had horrible consequences. So, the story works with the idea that Kaworu's explanation was the truth.
I'm not saying that Kaworu said absolutely no truth whatsoever. I'm also certainly not saying that he saw Shinji as just a tool, rather that according to this interpretation he ultimately would use him as one. Anyways, it just makes no sense for absolutely no one to bring up Third Impact even once in 2 movies, not even people who legitimately hate Shinji and would therefore be prone to portray him in the worst possible light and even overreact.
Who knows though, you might be right, in which case though the writing here is greatly impaired. There is no universe function we know of prior that could explain a supposed connection between N3I and 3I, so to have that single handedly explain an integral part of the conflict in these 2 movies, have Kaworu out of everyone else be the only one who tells Shinji the truth for some reason about this, and not explain said universe function a little bit or even bring it up again is just bad, jarring writing imo.
PS: I don't think Shinji will be against to use an Impact to repair the World, but I'm not even sure an Impact is really necessary for this. Shinji doesn't need to start a Sixth Impact (?) to repair the World in 3.0+1.0 (he just uses Eva-01 and the Spear of Gaius).
I actually think this interpretation is the most likely, that Additional Impact was cancelled by Shinji and that Gaius was used to physically pierce each Unit therefore making them disappear from that point onwards, instead of Shinji taking over Additional Impact and having his wish granted by it, while Gaius was used for said Impact like the previous Spears.
Even though Kaworu possibly omitting information about his plan isn't the main reason why I'm calling him a liar, I want to note that the world isn't really repaired in 3.0+1.0's finale (something Kaworu explicitly says he want to do), and it never seems like Kaworu's intention was ever to destroy the Evas. On the contrary, he always talks about fulfilling Shinji's "wish" and stuff, and well that's exactly what the Impacts do down to the specific wording according to Ritsuko during N3I in 2.0. I just can't think of what else Kaworu would want to do with 2 Spears and the Adam Unit 13 if not to cause another Impact, and I certainly don't think Shinji would just go with this and have no problem with this proposition after everything that happened.