I don't get why countries like Australia, Brazil, Canada, and the US get so much hate.
As others have said, it's mostly a case of simply not having exhausted better candidates. The US is obviously deserving of a spot due to its outsized influence on world history - a power needn't be long-lasting to have a major impact. That claim can't be made for the other three. Another issue is that the colonial civs mostly take areas of the world that could be represented by indigenous cultures - something there's a big community appetite for - and use Yet Another European LeaderTM.
Finally, by their nature these are late game civs and there simply isn't that much gameplay interest to them as a result due to the way Civ games work. So whatever you want out of Civ games' Civ selections - historical significance, diversity, or interesting gameplay - post-colonial civs are usually going to fall short in every category, and those do seem to be the three things the community most wants from civ selections. "Seeing my country" is a somewhat distant fourth, and the only real justification for using most of these civs.
Australia, it can be argued, also fills a TSL role that no other civ can, since there are cultural difficulties in portraying Aboriginal civs (specifically, a widespread - maybe universal - Aboriginal taboo on depictions of dead people, which both means there are few models of historical Aboriginal leaders to work from and could result in accusations of cultural insensitivity for using them). But it's a TSL role that isn't needed, since Australia isn't an especially attractive area to settle and sitting on a large island (in Civ terms) with no immediate competition doesn't make for great gameplay.
Yes, all their accomplishments happened in the past 200 years.
It seems that a greater portion of the community - at least if we're representative - values diversity than values civs for their accomplishments. And not to be unkind to a country where I used to live, but judged by a historical barometer it's not clear what accomplishments Australia has that really warrant inclusion, ditto Brazil. For two iterations now Civ designers have put Brazil in the game and appeared to have had no real idea what to do with it, so it's been a rather characterless culture civ with carnivals, football stadia and an ahistorical obsession with Great People. Australia comes off a little better (and has a great soundtrack and leader animation/lines that make it much more memorable than Pedro the Forgettable), but the characteristic chosen to represent it is .. settling in wilderness, farming cattle and getting attacked. Gameplay-wise it just gets excessive resource bonuses instead of doing anything interesting (though I'll grant that that's true of Nubia, a worthier civ on historical grounds). That's hardly on the same page as Sumeria, Rome or the British Empire.
These nations will also outlive every great empire in history.
What possible grounds are there for suggesting that? Brazil may already have hit its economic peak, having failed to meet expectations set in the '90s that it could become an economic superpower. Far right leadership intent on strip-mining its remaining natural resources is hardly likely to increase the country's prospects of surviving for the next few centuries.
Australia's global importance lies mostly in being Indonesia's bank and supplying China with raw materials, and it doesn't have the fertile land to support significant population growth - it already suffers severe problems from salination and water shortages in many parts of the country. It has an economic foundation in mineral extraction and agricultural products with an uncertain future market, so it's by no means sure that this is sustainable in the long run even without environmental challenges and its much-publicised and ongoing political problems.
Canada too is reliant, as others have mentioned, on harvesting and exporting natural resources. It likely has the brightest future of any of the three, but it's not clear why it should be considered in a better position than many long-established societies let alone empires that persisted for centuries.