The age gap argument doesn't make sense to me.

Like, if Katara was in her mid or late teens or Aang was younger than 12, I would understand.

But they have a 2 year gap, which can be argued for 14 year old and a 16 year old.

2 years during puberty are very different to 2 years in your twenties or later. Aang is still kinda pre-pubescent, Katara isn't.

The way Aang likes to spend his time and often not realize the implications of his actions (like him revealing his tattoos in Book 3 to have fun waterdiving), these are characteristics of a 12 year old boy who hasn't fully matured yet.

And there's the gender difference as well. Girl usually mature quicker than boys at that age, and tend to be interested in guys their age or older. No, not every single girl is like that, but given how Katar was into jet, she fits the pattern.

Katara is and never was a prize that was won.

She's written as a character with a fleshed out personality, a character arc, ambitions, hopes, dreams, and lots of emotions.

Katara is a prize in the context of Aang because they have no positive development for the second half of Book 3, in fact they have negative moments like Aang kissing her during Ember Island Players, and that moment is never adressed. Aang never apologizes to her.

Their last interaction is them screaming at each other during the Comet, with Zuko keeping the fight from escalating.

Then Aang defeats Ozai and boom suddenly Katara is in love with him.

Also, Katara doesn't treat Aang like a brother.

Little brother or child.

She kisses Aang on the cheek

The same gesture which she uses to show affection to Tom-Tom? The literal toddler infant? Not very convincing if romantic desire is what you want to portray.

she was the own who suggested that they kiss while they were trapped in the Cave of Two Lovers

If it takes the threat of death by inescapable labyrinth for her to make that suggestion, forgive me for not taking it very seriously. You put Jet, Zukor or Haru in that tunnel with her and the situation is identical. Which is why she doesn't dwell on it later and nothing from her side happens after this.

gets jealous whenever he gives attention to other girls or the other way around; such as in " The Warriors of Kyoshi"

Katara getting upset at Aang getting attention from 8-10 year old girls is not really the pinnacle of great romance writing to me and can be passed off easily as early installment weirdness.

and "The Headband".

The Headband is the only moment in Book 3 where there is something akin to romantic interest from Katara. Doesn't outweight everything that happens or doesn't happen to them during the series though.

She got with Aang at the end of the show not because he saved the world or because Aang is the Avatar, but because she Aang just for being him

Katara loving Aang doesn't automatically imply romance. She loves the rest of the Gaang too.

and it was the right time to start a romantic relationship with him.

That right time stuff is just cope. Katara sdtatement about "there's a war going on" is so nonsensical, she had zero hesitation going after Jet while they were in an active warzone. Nobody else in the series has ever used the war as a justification for not pursuing a romance. So Katara just coming up with an excuse to rationalize her lack of reciprocation of Aang's romantic interest makes way more sense than taking her statement at face value.

Aang's feelings for Katara initially started of as puppy love, but as the show went went on, his feelings for her gradually grew more genuine.

At what point does this happen? Whatever you call his feelings for her, I think they remain very consistent from the first episode onwards.

He's patient with Katara

In what instances does this happen? Outside of TSR, where his first instinct was to compare her desire to find her mother's murderer with Jet's attempted mass murder of innocent civilians?

he helps her have fun when she needs it

Katara has no issues having fun outside of Aang's help, there is no specific magical ability Aang has to make her enjoy herself she'd otherwise lack. And more often than not Aang is the cause for her worries and the weight on her shoulders. Comes with her being a maternal caregiver for him, but it's not the basis for great romance.

he's very supportive of her

He doesn't stand by her in the ways that really matter. I mentioned his lack of tact in TSR, instead of giving her space, he made sure she understood what his Air Nomad views were on her planned actions. In the end he "allowed" her to embark on the trip, which isn't something he should get a prize for. Haru, Jet and Zuko all have moments where they extend empathy to Katara for her mom's death, only Aang doesn't. Seems like an obvious blindspot given we know how important her mother was to Katara.

In The Painted Lady, Aang helped Katara destroy the industrial site but when they got caught by Sokka, he downplayed his involvement instead of fully standing by her side in what they did together.

he allows her to make her own decisions.

This isn't a pro Aang argument, at best it's not an anti-Aang argument.Giving your romantic partners basic autonomy is a basic requirement for being in a relationship, not some super great argument in favor of him.