What are those elements? There's a lot to say in answering that, but I can narrow it down to three major points:
1. "Dramedy," Not Comedy
It's hard to believe, but Ice Age was originally pitched to 20th Century Fox as an adult drama and the studio suggested changing it to a children's comedy. The end result was a hybrid of the two that the filmmakers have always referred to as a "dramedy." You can piece together the original plot from various sources, and while I think it was smart of the filmmakers to tone down or scrap a lot of those ideas, it was even smarter of them to keep the story's more tragic elements intact. The reveal of Manny the mammoth's backstory catches just about everyone off guard the first time they see Ice Age, and dramatic surprises like that make the film challenging as well as entertaining. Because of that, we get more emotionally invested in the characters and feel a much stronger sense of reward when they succeed in the end.
The sequels, on the other hand, are pretty much straight-up children's comedies. They have very few serious moments in them, most of the new characters have quirks and gimmicks in place of personalities, and the heroes usually joke their way through perilous situations instead of reacting with fear. What's more, since the tone is lighter, the characters' personal problems are a lot smaller and less engaging.
It's said that this change was because of Scrat, the sabre-toothed squirrel who's always chasing after his acorn in wacky, Looney Toons-style fashion. He was the most popular character from the first Ice Age, so the sequels decided to make everything in them more zany like his scenes. The problem with this is that Scrat works best as comic relief, and if there isn't enough drama for him to relieve us from, then he serves no purpose. Filling an entire movie with cartoony antics makes him less unique, and now his scenes interrupt the story in the sequels more than they offset it. Drifting away from "dramedy" was just a cheap idea with a weak outcome, and it hurt the comedic elements that were already in place.
2. A Touch of Humanity
The first Ice Age film is the only one in the series to feature humans--Neanderthals, to be exact. They're one of the more interesting elements in the film, not just because they give the story its emotional core, but because they never talk. This creates more of a contrast between them and the animal characters while also allowing scenes where everything is conveyed through actions and expressions rather than dialogue. Scenes like that can add a lot of depth and subtlety to a film, and they're especially nice to see in children's movies because of that.
I've read that the filmmakers did consider bringing humans back for the second film, but they decided to drop them and explore other aspects of the Ice Age universe instead. It's kind of ironic, because all that the sequels really do is introduce new animals who are more anthropomorphic than the ones in the first film. They'e basically humans, except they have more exotic character designs and they're allowed to talk. There's nothing wrong with featuring new prehistoric creatures in each movie, but when all of the characters are animals who can speak to each other, there's no sense of wonder or suspense in any of the encounters. We know exactly where the heroes stand with everyone because everyone's motivation is said outright, and the intrigue of connecting with other species is mostly forgotten.
The closest the sequels ever come to recreating that dynamic with the humans is with the non-talking dinosaurs in the third film. Even then though, the emotion is somewhat lacking because the dinosaurs are too out of place in that time period. If the filmmakers wanted to explore other aspects of the series's universe, humans could have still offered a lot of different ideas. There were other species of cavemen living at the same time as Neanderthals, and cavemen did cause a lot of problems such as overhunting and competition for food and territory as their populations grew. However, more talking animals in the sequels meant more opportunities to cast big-name celebrities, so spectacle won out over heart once again.
3. Less is More
The first Ice Age movie was about three animals returning a lost human baby to its family while bonding with each other and trying to survive a pack of predators. That's it. There were no major global disasters, no discoveries of lost civilizations, and no magical, outer space excursions. The story was small and simple, and that gave it enough time to properly flesh out its most important elements.
There's too much of everything in the sequels. Too many characters, too many subplots, too many twists and turns, and too much at stake. It's overwhelming, and with so much crammed into each installment, none of the story elements has enough time devoted to it. There are interesting elements in the sequels, such as the character Buck in the third and fifth films and the relationship between Diego and Shira in the fourth film, but they only get superficial development. Imagine getting a flashback of what happened to Buck's family and how he came to live underground with the dinosaurs instead of just seeing a goofy story about how he lost his eye. Imagine getting a flashback of why Shira left her old pack of sabre-tooths and being reminded of what Diego's pack from the first film was like. These subplots could have been really compelling if they didn't have to share their hour-and-a-half run times with a dozen other stories.
Another issue with putting too much into the sequels is that it strains their plausibility. Ice Age: Collision Course is about averting an asteroid by launching magnetic rocks into space from a volcano while a squirrel in a spaceship plays billiards with the solar system. Not only is it nearly impossible to take that main premise seriously, it's also difficult to take the characters' personal story arcs seriously because of how ludicrous the threat to them is. I know that the first movie had a few implausible moments too, like the ice cave full of dinosaurs and spaceships and the river of lava that erupts out of nowhere, but at least those scenes were short and barely referenced again. Heck, the latter example actually sets up one of that movie's most earnest character moments, and since the lava eruption was likely caused by underground volcanoes which were common in prehistoric times, the character moment is able to overshadow the disaster. Again, less is more.
We're already hearing rumors that a sixth Ice Age movie will be in the works soon. If that's true, then I definitely think it should be the final installment in the series and that it should bring things full circle by reintroducing the above three points. I think it should be the film that finally brings back the baby from the original as an adult, and just as the characters speculated in the first film, he should now be a hunter who doesn't remember being rescued by animals when he was little. Have the character who first brought Manny out of his depression over his first family now pose a threat to his new family. That would be an interesting, dramatic conflict with a lot of tension and uncertainty, and the main premise could involve the humans and animals clashing with each other after a disaster at the start of the film forces them together. That way, we could still get some of the larger scale antics of the sequels while also having the drama of the original.
More likely than not though, this won't be the premise of any future movie. We'll probably get another sequel or two much like the last four until the franchise bottoms out, and then Ice Age will go on the same shelf as The Land Before Time and Home Alone as a good movie that spawned a baffling series. In the end though, that one good movie is all that matters. Hollywood could hurl a million sequels down on us that crash and burn, but however big their craters are, we can always find the original with all of its simple, "dramedic" humanity perfectly preserved in the ice below.