Avatar: The Last Airbender

It does usually contribute to the story but it is filler

Honestly I don't agree. With the sole exception of The Great Divide, I think every episode in the season adds something important. There might be a little room to tighten it up, but it would be hard to remove entirely without losing something.

The first two episodes do so many important things. You could maybe tighten it up a little, but not much. You need to establish that Aang is both a fun happy-go-lucky kid, but also one who is aware the weight of the world is on his shoulders. You need to establish Zuko's inner conflict, and the dynamic of Sokka and Katara. And, obviously, set up the wider world, establish the Fire Nation as baddies, the Avatar as powerful, the nature of bending, etc.

The Southern Air Temple hammers home the Air Nation genocide's significance, and as well as establishing the Avatar's reincarnation cycle.

The Warriors of Kyoshi is a comparatively more filler type episode, but it's important in Sokka's growth in terms of the obvious growing-out-of-sexism arc, but also in showing how non-benders (including him, as well as the Kyoshi warriors—who become more important later) are able to put up a serious fight. The destruction left by Zuko helps hammer home how ruthless the Fire Nation is in a more immediate and direct manner than the previous episode's allusion to events of a century ago.

The King of Omashu probably the second-most skippable of the season. But it establishes King Bumi, who obviously becomes important later both as a member of the White Lotus and in terms of his talk about neutral jing, and it establishes the first major city that we see, and one which is important later when it gets captured by the Fire Nation—a beat that is a lot less powerful if we don't already have a good connection to it.

Imprisoned is important because it gives us a brilliant performance by George Takei. 'nuff said. No but seriously, this is the first episode since the pilot that gives us a deep focus on Katara and her more justice-focused personality than the other characters. It's also our first serious look at the colonial nature of the new Fire Nation empire. Less importantly, it's collecting our first character who is later involved in the Day of Black Sun invasion.

Winter Solstice Part 1 establishes the rules for the Spirit World and the Tolkienesque "importance and power of nature" theme. Part 2 is much more important in its direct plot significance.

The Waterbending Scroll gives us: the first hint of the White Lotus, the pirates who will later be used to blow up Zuko's ship, the start of the very personal animosity between Zuko and Katara (an important part of both their character arcs), a demonstration of Aang's natural aptitude and Katara's work ethic, evidence of the two of them studying at waterbending.

Jet is another relatively skippable one, except that it shows us for the first time how there are evils on both sides, even if the evils on one side far outweigh the other. An important theme with Ba Sing Se in season 2.

Then there's The Great Divide, which we'll skip for obvious reasons, followed by The Storm, which I cannot believe isn't the episode you listed as the one where "it really hits its stride". It not only provides the necessary set-up for The Blue Spirit, but it gives us the critical backstory to understanding who Aang and, much more importantly, Zuko, are. It shows us Aang's heroic flaw, and the spark that ends up leading to Zuko's redemption. (Also, weird side note: this Wikipedia page about character flaws honestly reads more like a TV Tropes article, replete with dozens of specific examples from different media, ranging from Macbeth to Aang to an entire section on the show Once Upon a Time.)

I was originally going to do the whole season, but this has already gotten long enough, and I've caught up with where you said it finally lands, so I'll stop.

Am I saying that the adaptation is automatically going to fail because it's leaving some of these out? No, absolutely not. In fact I think for the way most TV is told these days they're probably making the right decision. More highly-serialised longer episodes are kind of by definition going to necessitate certain more fundamental structural changes, and collapsing many of these arcs and themes into side plots within the same episode. So no, I'm not criticising the live action showrunners based on what we've seen so far.

But I am saying that the original show does a much better job than I think you give it credit for of using its episodes well, and that cutting much out of even the first season would be difficult in the context of a 22-minute episodic cartoon. In fact I think Avatar is perhaps the most perfect example there is of a TV show that uses episodic storytelling to its fullest potential. Each episode is fully capable of standing on its own (something most more serialised shows cannot claim), while also forming a crucial part of the greater whole.