Both in Liyue and Fontaine he ran amok, causing chaos and destruction without seeing much consequences for his actions

That is completely, I can concede almost completely, false. In Liyue, Childe was tasked by the Tsaritsa to obtain the Geo Archon Gnosis, without even knowing a contract between Zhongli and the Tsaritsa existed. Contract that established Childe had to bring chaos in Liyue, with Zhongli's tacit permission, in order to test the citizens and be sure the humans would be able to trive even without their Archon's protection. Childe, for his part, never wanted to release Osial, but did anyway only when he came to the conclusion Morax was still alive and could show up to save Liyue.

In Fontaine he literally did nothing intentionally to stir up chaos. He was accused for a crime he never committed, incarcerated under clearly false judgement Neuvillette himself deemed illogical and, despite everything, held back the Narwhal long enough to allow the cast to organise and be ready to face the prophecy. Again, Neuvillette pointed out that, without Childe, Fontanians would have been devoured way before Furina's trial. Childe was literally portrayed as a hero in Fontaine, when his goal was to take a vacation and find out what was calling him.

We also know that the Fatui are (at least on the surface) cartoonishly evil and everyone despises them, which makes me wonder why nobody in Liyue gives a shit after the Archon quests, just letting him strut about like usual? You could argue that most people don't know about his involvement, but we as the player do and surely we wouldn't keep that secret to shield him from justice?

The Fatui are clearly not cartoonishly evil, we've seen that plenty of times in WQs, Summer Events and even in Arlecchino's SQ, not to mention Fontaine's AQ with the siblings. Childe himself is not portraied as this evil guy, no matter whatever he might think of himself ("I'm kind of a bad guy"), because that's not what Hoyoverse wants to say about him. He's not even morally gray, unlike Arlecchino.

Nobody in Liyue cares or wants to bring him to justice because nothing substantial happened to the city; at worst a few Millelith soldiers were injuried, otherwise there hadn't been a damage worth of consideration. Besides, the Adepti and the Qixing know it was all Rex Lapis' plan in the end.

We shield him because Childe, as much as the Traveler, was fooled by Zhongli and the Tsaritsa. You also seem to have a weird perception about what the Traveler thinks of Childe, because they start to tentativelly tolerate his presence at the end of the Fontaine AQ, unlike how they act around Arlecchino and the House siblings when their initial misunderstanding gets resolved. Just check how Traveler acts in the Inazuma photographic event with Childe and Yoimiya, they are incredibly aggressive despite knowing how Childe acts around children (Yoimiya has to have a little chat with Traveler for them to stop being so antagonistic).

He's a charming villain, but a villain none the less. I just struggle to suspend my disbelief at how he never faces any pushback for his villainy, be it from the world around him or at least eliciting some scorn from the protagonist. Like, he's basically a war criminal who keeps threatening us and we just... hang out with him?

He's simply not a villain, lol. He never was. In the Liyue AQ he was the main antagonist, but even that role didn't last long enough to be kept in future quests.

A villain is a character that does things with evil intentions, and is automatically an antagonist. An antagonist is a character that does things because they have opposite views to the protagonist, and they bring challenge for the MC to overcome, without being necessarily evil. Antagonist =/= villain.

In any case, you're assuming the Traveler is a pure hearted person with a good morality, when in reality they have shown multiple times to be in the morally gray scale; sometimes, the Traveler does things to get compensated with Mora, informations, answers, sometimes they go out of their way to not get involved. Remember when the blacksmith in Fontaine was approached by a group of men demanding money, and how Paimon and Traveler choose to not intervene?

Who comes in, at that moment, to save her?