A stablecoin issuer that makes claims about converting the tokens for dollars is a money transmitter (Circle), in that like a bank they receive deposits/reserves (actual dollars). A bank receives payment instructions through checks/etc to send 'electronic money' to someone else, based on real dollar deposits. A stablecoin issuer that makes claims to token holders that they can exchange the token for a dollar is doing the same thing as a bank just a different way- the way 'electronic money' is sent is through the token on a blockhain, p2p. They do not receive payment instructions, but they do custody the dollars that can be converted on presentation of tokens. But because they issued this token, that can be moved p2p, this token represents a claim on those dollars sitting at the stablecoin issuer. So the token/blockchain is moving funds electronically.
Other cryptos do not work this way, since they aren't convertible to anything. I would say any issuer of any token that makes claims to exchange some type of reserve (another crypto) is also a money transmitter by the same logic.
Tether says it doesn't have to exchange tokens for dollars legally, but that doesn't mean that they don't ever. If there is no legal claim on the reserves for token holders then Tether isn't responsible for money transmission when tether is sent p2p. Tether is a money transmitter though when they sell tether for dollars, because that is a business.
Cirlce (USDC) issuer is a money transmitter, but they can only do KYC/AML directly on the front (who they sell to) and on the back (who presents token back for dollars). They can't do KYC/AML in the middle, where people can do p2p hops. Even though these people are not getting 'real dollars', the token itself is the technology that allows for money transmission and represents dollars held. What about cash in the bank? When you withdraw cash you give it to anyone and it can be accepted and there is no KYC/AML. The difference is there is no counterparty to cash dollars. They are real, they are money, you can hold them in your hand. They are different than your electronic account balance at the bank, because it is virtual (a representation) and because it is a liability of the bank to you. Just because USDC (the representation) can be withdrawn into your control- meaning you do not need Circle's approval to send a payment p2p (although they control the token so do have some authority), that is not the same thing as withdrawing cash dollars. USDC is representative of cash dollars, not cash dollars themselves, thus do not have the same properties as dollars. Dollars (cash) is real money with no counter-party risk. You can give it to anyone and the transaction is settled immediately. Banks hold financial assets, financial assets that are intangible always have counter-party risk. Someone who has USDC can send it to anyone too, but it hasn't been settled, because the dollars are sitting at Circle, the issuer. Because the real dollars are always sitting with the issuer, they act just like a bank where anytime you move money from your account to someone else electronically, everyone, everytime is KYC'd/AML'd.
Confusion is around separating the technology employed to move real money, which can be called electronic money, and real money. Stablecoins are just an a new technology to move electronic money, not real money. In fact, all cryptos are a new technology to move representations of something, usually dollars/currency (if it is has a price), but they are not real money themselves.
There is no way to kyc/aml p2p crypto hops for a central stablecoin issuer, thus violating their own money transmission requirements and this is because the tokens are mere representations of real dollars, they have counter-party risk, they are a financial asset under the issuer's control.
Even with additional regulations, they can never mold this behavior of the nature of p2p tokens under existing money transmission rules. It seems to me eventually unknown p2p transactions will be banned, and all public addresses will have to be identified by Circle, or else they will just 'delete' those coins, so it can be done, and if it can be done, it will be done. Stablecoins are only beneficial if that does not happen. Tether is your best hope of keeping status quo.