I don't try to recommend anime for the sake of it. I try and think of what someone likes and their threshold for certain things.
My mother likes John Woo movies and Law and Order. I was able to convince her to watch Cowboy Bebop and Stand Alone Complex 2nd Gig. She legitimately liked them. I'm not going to push Gurren Lagann on her because it's possibly not in her purview. Things with understandable action and cop dramas are.
I kind of threw a bunch of anime at Shadow, but it sounds like shes looking for things that are mature, well told, and lacking in any sort of otaku bait.
I'm kind of pushing with Giant Robo, but I feel it is a fantastic and criminally underrated show. Over the top, overly dramatic yet operatic, sincearly heartfelt, and extremley well plotted for a art decoish Wuxia robot anime that I wish more people would see.
Macross Plus is just good olde fun about a pilot riviarly. It has aliens and other stuff, but the actual plot is not something that goes over your head. It's not something form a young adult novel where the world is crazy if you think of one aspect of society and go the deep end like so many modern anime plots feel.
The rest of things I recommend to people who aren't knee deep into anime are things I dont think would scare people. Things I think would basically work as actual live action movies. Things not too far removed from what you could find in mainstream Hollywood.
Patlabor 2 is a political thriller, not far removed from your post 9/11 thrillers.
Sword of the Stranger is on the order of John Wick.
and so on.
