Actual dollars can't be traded as easily on crypto exchanges, it's as simple as that. USDT (Tether) is traded just like any other token riding on top of a blockchain. If you wanted to hold USDT and didn't trust the exchange, you can withdraw it to a wallet you control, just like any other token. (Most crypto nerds don't hold USDT, though, they use it to trade between tokens which they then store in their wallets.)
If an exchange tries to keep it's own dollar-denominated accounts, then their customers need to trust them to not run off with those deposits. It would also subject that exchange to some amount of US oversight, particularly if they also want to participate in the US banking system and let customers deposit and withdraw via ACH.