The part in bold is not true. There are some free tools and there are many paid tools with which you can check if the coins are "tainted". Putting aside our ideological disagreement with the tainted bullshit, these tools will give you exactly what an receiving institution or exchange is going to see when they receive said coin.
How do you know that an exchange uses those tools? Maybe they do, maybe they don't and have some other tools.
Tools that are industry standard rely on the same data sources, they don't have to use the same exact tool to see the exact same information -- If this was not the case, there would be complete chaos and it would be near impossible to use multiple reputable exchanges at the same time in any sane way. Furthermore, experimental evidence will confirm it but you have to do it yourself. I have tried with countless transactions using clean coins, and sometimes with tainted coins to see what would happen down the road with one of these systems. If you don't believe in the theory/research, then you can do the tests yourself with some tools assuming that you have money that you could burn as an expense to verify it for yourself.
Not sure what you mean by "wallet accounts". Im mostly familiar with Bitcoin Core and Electrum.
Coin Control without accounts is for noobs, and mostly old timers that have outdated methods and knowledge tend to suggest people things like that. A single seed can derive countless addresses in countless accounts, so you would have something like this:
seed - coin - account 0
seed - coin - account 1
By using them this way, you can ensure that there is no leaking between your accounts. For example, account 0 can be your bigger hot-wallet stash. Account 1 can be restaurant payments. If you use accounts correctly no error that you commit within a single account will leak anything about the rest of what you have going on in the same wallet. Whereas with traditional single-account coin control, all you need is a single basic mistake to undo a lot of work and reveal a big portion of your history. Accounts are pretty easy to do in Sparrow:
https://www.sparrowwallet.com/docs/connected-wallet.html#bonus-adding-an-account. Try it, you will understand best by trying it.
As far as atomic swaps, are those the same as "no-KYC swap services"? are those the same as "aggregators"? It's confusing differentiating all of these.
No, they are not. They are completely different services. Some classes that are relevant:
1) Atomic swaps, for example those that serve XMR <-> BTC. Here the limits are technical, and liquidity. You are not going to find various routes for different coins or tokens, depending on what you have and need. However, you get the most anonymity here.
2) Decentralized bridges, for example those related to THORchain. Here the limits are related to anonymity, a technologically adapt observer can trace your routes. The benefit here is that it can not be stuck for AML/KYC reasons, and it would auto refund.
3) "No-KYC" swap services such as ChangeNOW. The downside to this is that it is centralized compared to the first two. In most cases you will not have any issues, but if you are unlucky or trigger the system in some way you will be forced to do AML/KYC or abandon the coins that you are currently swapping. Upside here: Only the service can de-anonymize you, and the coins that you receive from them are AML friendly.
Strongest anonymity: Atomic swaps.
Decentralized with many routes: Decentralized bridges.
AML friendly coins with most routes: Swap services.
That is how I would rate them in general. Please keep in mind the risks and disadvantages of the options, and use what you personally need -- don't listen to the biased idiots on this forum. I have used hundreds of services while testing systems in the last few years, I have very rarely encountered issues with anything and even in those cases there are many things that you can do. Experience is key, therefore pick something and start practicing with small amounts.
Anyway, nonetheless im interested in what AML checking services do you know, because like I said, from what I have read, the powerful AML checkers are reserved to professionals, so you would need to pay a lawyer/tax guy to check your addresses out if you want a real peace of mind, compared to some free AML checkers which we also don't really know how they work or who is behind. However doxing yourself to pay some guy to check your addresses on that professional software has risks in itself. What if you hold risky AML coins and they show up? It's a bit of a slippery slope, that is why I never saw a clear way to deal with crypto payments and been treating it mostly as a store of value and barely moving any funds.
I would not recommend doxxing yourself to some "guy" to do an AML check for you,
it brings no advantages at all nor does it bring any guarantees of anything. This kind of service is more for the people that have not grown out of the thinking in "authorities" way, you know the kind of people that would ask their doctor how they should drink a glass of water.

Here are some right now, keep in mind that things change over the years (and I hope that I pasted the right links):
https://amlbot.comhttps://www.scorechain.com/resources/crypto-glossary/aml-wallet-check-freeThe best solution for this kind of topic for those that really need it is to be in the network of someone who has access to Chainalysis. I've had the best success with that, I don't remember a single case of coins that were graded as low risk by Chainalysis having issues later. Of course, the biggest obstacle here is getting access to Chainalysis.