So, I watched Evangelion but basically none of the others you mentioned. I feel like such a fraud as a moderator of !gundam@ani.social, but in any case, I can recommend at least one mecha series that I particularly enjoyed. It is more a parody of the genre, but sometimes its through parody that we realize the best that a genre has to offer. Specifically, Brave Bang Bravern!.

Is Bravern a comedy? Yes. Is Bravern a tragedy? Yes. Is Bravern a romance? Yes. Is Bravern kick ass? Hell yes. It’s hard to put it in a box since it will pull some new weapon out of nowhere and break out as soon as you try. At one point, the giant, sentient robot was building giant robot-sized gunpla, and this was just accepted as normal and never commented on by any of the characters. It’s an absurd take on the mecha genre that simultaneously shows an absolute love and dedication to what makes it great. The genre might be about big robots, but it is the human connection that makes us care about the story at the end of the day.

tl;dr - Watch Bravern