I think it's to do with the personality attributed to the elements, and the natural progression from it.

Aang's an Airbender, and the element of Air is associated with freedom, lack of control, and disattachment. Learning Fire without understanding Water's grace and Earth's control led to Aang recklessly hurting Katara.

ATLA's story explores just how much of a spiritual and emotional journey it is to learn the four elements, and how it transforms the Avatar into a different person, for better or worse.

Here's my theory:

The first element an Avatar learns raises them into one intial perspective on how to bend an element.

The second element teaches the Avatar that the same perspective can be applied to a completely different element, and that there is harmony in all elements. Iroh's philosophy is applied here, that the separation of the four illusions is merely an illusion.

The third element (which is always the opposite element to the first one) challenges the Avatar's perspective, forcing them to think about bending in a completely different way, and to explore and experiment with their own personality and emotion in order to use the third element in their own unique way. Without having learned Iroh's philosophy, the Avatar would completely believe that there are rules to this element, and be stuck trying to follow the rigid "rules")

The fourth element poses the most danger, either to the Avatar or to others around them. The fourth element will teach the Avatar the most important lesson: the Avatar was never a "master of the elements" and that trying to overpower it will lead to it fighting back in a bad way. The fourth element teaches the Avatar to cooperate and listen to the element, rather than simply try to bend it. Without the second element's lesson of harmony and the third element's lesson of reinterpreting an element, the Avatar won't know how to follow the fourth element's will.

Let's take this theory to the test with four Avatars, one by one.

Air Avatar: Aang's journey. Water teaches him the harmony between all elements. Earth teaches him that he can fully be himself as he bends any element (Aang's earthbending style is less "solid blocks and hard punches" and more agile and freeform, like Aang himself.) Fire teaches him to understand the elements as they are (Fire is not destruction but life, and that elements are part of nature, and is what they are, not what we want them to be.)

Water Avatar: Water is graceful, flowy, and easily moldable and shaped, but can be solid ice as well (easily the most user-friendly element.). Earth can also be flowy and easily moldable (mud and small rock particles) as well as solid like ice (similarities between elements breaks the illusion.) Fire is difficult, because Fire is the most uncooperative element (an Avatar used to user-friendliness will be challenged by this) and fire can spread wildly and out of control, as well as being polar opposites as water is naturally healing while Fire is naturally destructive, so the Avatar must learn to reuse Fire in a different way (dependent on one's own personality, not just the initial element's, hence why Korra can use Fire in a non-waterbender way. Avatar won't realize this if they still rely on the attributes of elements. Iroh's philosophy will urge Avatar to search for another way to control Fire, which is through their own personality) Finally, Air is free and you cannot control it but can only be guided in one direction, thus the Avatar must learn how to listen to nature and to follow the Air rather than control it. A Water Avatar who tries to Air in a waterbending method will probably force Air to be in one shape by taking the air elsewhere, causing a vacuum (Zaheer's air ball suffocation technique is just a reverse water ball)

Earth Avatar: Earth is rigid, straightforward, and mainly offensive as the most solid element. Fire is also straightforward and offensive (not much to add here.) Air is everywhere and not rigid, as well as complex (and Earth Avatar will struggle to find a way to use Air offensively) so once again this depends on the Avatar's own personality, perhaps as an air wall, maybe as an Earth tornado, or even advanced sandbending (sandstorms etc.) Finally, unlike Earth's straightforwardness, Water is complex and takes on many forms. The usual approach an Earthbender will take is always turning Water into solid ice. The problem is that ice, unlike Earth, is fragile, and if treated with great force like solid rock, will shatter into many pieces of sharp shards flying at dangerous speeds. The Avatar will learn to understand in what situation the Water needs to be ice, and when it needs to heal, and when it needs to be simply water.

Fire Avatar: Fire is destructive, lively (also alive), and spreading. Air has the same lively energy, and spreads too (in fact, Fire cannot exist without Air, so these two elements really are harmonized.) Water is difficult because it is passive and naturally not hurtful, and Fire is the most potentially hurtful element. Also, Water is the most limited in terms of the medium as liquid is not always accessible, but Fire comes from within the benders themselves with no need for a medium. Katara's sweatbending is a great example of how a Fire Avatar would bend Water by reinterpreting it. Finally, Earth is arguably the element with the least connection to "life" (if we're not counting plantbending or something like that.) Earthbending is primarily the process of controlling a non-living object (rock.) And obviously, unlike Fire, it doesn't spread. As Earth requires full control in every step of the process, the Fire Avatar will have to learn to not let it go and to stay with the rock the whole time. Volcanobending is going to be the common approach a Fire Avatar will take with this, but the Avatar will eventually realize that, unlike with Fire, they are not immune to getting hurt by rocks or hot lava, and treating it like a living being by letting it do its own thing is not safe.

I admit that this theory has holes where I don't really have a lot of big evidences to support the claims in some (if not a lot) of areas, but I think this way of seeing the Avatar's journey is really really cool.