Bitcoin does not have swap. Swap is for altcoins.
You can exchange your bitcoin with peer to peer trade partners or trade it on centralized exchanges and withdraw it to your non custodial wallet which can be different with a wallet you used to fund your exchange account.
You can use mixers.
You can use Coinjoin for your Bitcoin transactions, from your Bitcoin inputs to Coinjoin pool and receive bitcoins in Coinjoin outputs.

If you are careful with privacy, you must stack your bitcoin privately, it's first important practice. Then store it in non custodial wallets, use change addresses, choose UTXOs carefully for your transaction inputs, and use Tor.

If you have these steps practically bad, no-KYC exchangers are helpful but can not erase your information on chain.

You can not swap your bitcoin to shitcoins. If you want to swap like this, you must have your bitcoin converted to Wrapped Bitcoin tokens before swapping those tokens to shitcoins on different blockchains.

Wrapped tokens are not bitcoins, and many people are both confused about their difference, as well as ignore risk of Wrapped Bitcoin tokens. Those tokens are on altcoin blockchains so they can be affected by weakness and security of altcoin blockchains, and they might lose their token pegs with Bitcoin price on the market.

Definition of tokens.

They can find such no KYC exchanges with some sites.
https://bitlist.co/exchanges
https://orangefren.com/
https://bitmixlist.org/

There are sites for checking AML scores too but I don't use them because if I care about privacy, I don't trust those sites in keeping up their words and don't break my privacy.

If people are not careful when checking privacy scores, AML scores, they can lose their privacy similarly to how to use Bitcoin block explorers.

Why is it that every time we talk about privacy AML has to be mentioned, isn't it that privacy is not just for money laundering? So using an exchange to swap our Bitcoin to altcoins is just a way of keeping our privacy, so it should not automatically trigger an investigation or be linked to AML since we are not committing a crime or laundering money?

No-KYC has nothing to do with protecting your privacy as in your IP address or the transaction history, it only allows you to use their services without providing any personal information. However, generally, some platforms do say that they only keep IP logs and such stuff for a limited time, and then they are removed which means that you basically don't have anything to worry if you are using such swaps and services and not centralized ones.

When it comes to general privacy or blockchain history, then that isn't really that big of a problem as long as you don't have your identity attached with your wallet due to transactions you have made in the past. If you have a wallet and also have a centralized platform account, and you have sent and received assets between those two, then your wallet can get identified as your wallet if someone investigates about it, but if you haven't done anything like that and have only used decentralized and no-KYC platforms, then you have nothing to worry about.

Transaction records in the blockchain are important, if there were no records at all, things would be worse, and that wouldn't just be a privacy concern but we wouldn't have enough information about our transfers, etc.

You have to tell that to where you’re moving the money to..  . 

Depending on the type of privacy you’re trying to achieve.. when it comes to anonymity, money  launderers and criminals do same,  so the receiving end will most likely categorize some transactions as one  and that’s even if you don’t end up with a tainted coin in the process . But just as I’ve said, it depends on what privacy you’re trying to achieve.

If you have any means to verify further please do without hesitation and I think my recent post was almost taking the issue with the lapses we are facing current with bitcoin decentralization and privacy and since we started embracing government into our system, we started losing the decentralization and privacy we truly cherished and it is saddening that there's nothing we can totally do about it because even if one deceives to use those platforms that claim to be decentralized can't just be trusted considering the level of scam and fraud in the industry and it is always very risky and that is why the people mostly have to fall back to these centralized exchanges because it they offer some level of security while trading but that security is what we end up trading our privacy and decentralization with.

I guess using TOR/VPN is a good step in the right direction first for true privacy because a no-kyc label is just a label that is more as a marketing gimmick than as a true quality of privacy.
There's also the part of checking the open source logic and data minimization to ensure absolute privacy because privacy is a serious business if you are a true and devoted investor and don't rely on exchanges or their settings to see your coins and transactions safe in the end.

Most of the questions are unfortunately not possible to answer, for users:

This is the only one you can actually check, but maybe the least important one. Even if a platform requires registration, it could be managed in a way that the individual swaps aren't stored, and thus privacy could be preserved.

If the platform has a proper ToS then you can check what it claims to collect, log and store. But you can never be sure that it's true. The strongest way to ensure this is maybe that the platform gives a legally binding guarantee in their ToS that they don't store the data. But a simple "We don't store X" is not enough to really prove that.

This is often used in a misleading way.

If the swap passes through addresses owned by the platform operator, then it's custodial. Even if the funds are only a millisecond on the exchange's address before the payout is made. The exchange could go offline in exactly that moment.

The only non-custodial solution between different blockchains is atomic swaps. Even THORchain is custodial, however there are strong incentives for the validators not to misbehave.

I don't question you're probably a reputable platform, but complete trust is not possible. There is no way to prove that the exchange hasn't logged or stored data.