>Especially compared to post-millennial cartoons which have been quite sterile
OP you couldn't be more wrong about this even if you tried. My friend let me kindly introduce you to a mini-series made by Cartoon Network, Over the Garden Wall availble on itunes and amazon. Sorry if I seem to be shilling out but this show deserves every praise it gets.
Let me summarize the show in three words,"American Spirited Away". Check out their voice stars, Elijah wood(first spider man), Christopher Loyd, Jack Jones, Samuel Ramey, etc. here
Now back to topic as stated on hand as stated by /u/agentxorange127 Hey Arnold has been a good cartoon and always will be. But stating his terms on nostalgia I agree that you are cherry picking from the episodes that were specifically for the characters to develop themselves, such as pigeon man, the Christmas special, valentines day special iirc some of the best episodes of the show.
However it is not the only cartoon in past nor recent years to have heartwarming and thought provoking episodes. Courage the cowardly dog, Recess, Adventure time, Gravity Falls are only some of the cartoons that had these special moments. I believe what you are looking is cartoons that do not look down on children's intelligence and respect them as actual people with thoughts and feelings.
One of the main criticisms of the show was how Arnold changed from a regular everyday kid to a wet blanket voice of reason. Now this may be attributed to the fact that he has no parents and this must grow up faster. But the drastic change with no explanation when iirc Season 1 Arnold and Season 2 Arnold are vastly different. The Arnold then was still kind but iirc more mischievous while the later seasons had Arnold shooting down his friends fun for various reasons. It is a good show however marred by stereotypical imperfections like other cartoon shows for kids like Avatar: The last Airbender, Danny Phantom, Kim Possible etc.