I'd argue that the main reason you see more anime is the target audience.

Western animation is usually aimed at young children. For as much as I may have loved Disney's Gummi Bears as a young child (decades later and I can still hear the theme song on my head), it's now pretty painful to watch. Some shows have aged pretty well and some newer shows aren't quite so bad. But, the target audience still seems to be younger children for much of it. There are exceptions, and several of those are pretty well known. For example, The Simpsons and Futurama are both popular animated shows, and both are not aimed at children.

Anime, by contrast is often aimed at teenagers. This means that it's part of the audience's formative years. People form bonds with the shows and carry some of those bonds into adulthood. And while the writing often falls into cringe inducing melodrama, there's enough of it that is passable fun, usually simple hero stories. The shows can be like a comfy blanket that doesn't insult the audience's intelligence too much.

I'd also note that anime's appeal goes back further than the 2000's. My own introduction was Robotech, back in the 80's. While it was a bastardized version of Macross, with some pretty awful writing (not that Macross's writing is going to win awards any time soon) and a couple other shows, it was certainly a step above what most western studios were putting on for Saturday Morning cartoons. And that created a lifelong soft spot for anime. Heck, my desktop background is currently a Veritech Fighter. I still love the idea of Robotech, even if I only watch it in my memory through very heavily rose tinted glasses. And I imagine I'm not alone. The show may be different, but I suspect a lot of folks graduated from Disney and Hanna-Barbera cartoons to some type of anime as they got older and that anime was stuck with them.