If I recall correctly, Lain was fairly popular both in the West and Japan, largely for similar reasons, although it was less big in the West since at the time it came out anime had not really become A Thing yet.
To your point, Cowboy Bebop can probably be considered to be one of the biggest causes for anime to gain so much traction in the US due to being on Adult Swim and having a killer feel and soundtrack, but its lasting power wasn't nearly as strong in Japan.
IIRC Cowboy Bebop was developed explicitly with western audiences in mind. It’s success informed later shows (in a similar way to how thr Chinese market is influencing Hollywood movies) so it really is a landmark show, even if it wasn’t as well received in Japan.
I mean they made a Movie spinoff from the show, so it was sufficiently popular at the time. It didn't have mass appeal staying power a la Eva or Geass, but the former is an outlier and the latter was made in an era where the production companies became much more sophisticated about fan cultivation and pre planned media mixing.
Oh yeah, I don't mean to suggest it wasn't popular, but it also doesn't seem to have had the same level of sustained popularity that it has here in the West.
To your point, Cowboy Bebop can probably be considered to be one of the biggest causes for anime to gain so much traction in the US due to being on Adult Swim and having a killer feel and soundtrack, but its lasting power wasn't nearly as strong in Japan.