Also, if you want to move money between exchanges or banks you'd have to use the regular transfer methods like ACH, SWIFT, SEPA, which oftentimes are not as fast and cheap as sending USDT between exchanges. That makes it much harder to exploit arbitrage opportunities and drives up the spread overall, for example.
This is the biggest reason, really: the moment you touch USD you have to comply with know your customer laws, review OFAC sanctions lists, etc. And this doesn't just apply to US-based companies, if you are sending money through the US financial system or handling dollars, this applies.
I'd recommend reading https://www.kalzumeus.com/2019/10/28/tether-and-bitfinex/ (Tether: The Story So Far; by Patrick McKenzie aka patio11)
Basically, the US requires money-transmitting entities to know who they're working with (https://en.wikipedia.org/wiki/Know_your_customer). When you open a bank account, the bank has to validate your ID. Even if you're opening it online, they have systems that ask you questions to validate your ID.
If you're trying to create a non-tracked/anonymous system, you ultimately lose access to the US banking system (and probably the banking systems of a lot of other rich countries).
I'd say that it's not just "heavier regulation". That makes it sound like it's just hoops to jump through or delays in the processing. A big part of it is that banks/money transmitters/fintech need to know who their customers are in a lot of countries. It's not just random stuff like, "they need to fill out these forms, file these reports each week, etc." If you're trying to run an anonymous exchange and possibly profit off money laundering, anonymously transmitting payments for prohibited activities, etc. you don't want a system that requires you to know your customers. You just want to be, "it's crypto - things come in, things go out and it's all anonymous!"
There are companies like Coinbase which are licensed in the US, but Coinbase probably isn't where you'd want to go if you were looking to launder money. If you're looking to launder money, you don't want to go with a company that has been working with regulators. It would be relatively easy for Coinbase to give you USD, but Coinbase also knows who you are and would comply with warrants.
If you're a crypto company that doesn't want to comply with authorities, know your customers, etc. then it might be hard for you to give someone USD. However, it's really easy to give someone USDT - you just put it in your ledger. I can keep a sheet of paper and say "I owe X 100 MyDollars" and say "Every my dollar is backed by a USD in some form". Even if I'm 100% telling the truth, it doesn't mean that I can give you the USD I have in my pocked if you give me a MyDollar. Maybe you don't live near me so I can't give it to you. If I'm cut off from the banking system, I can't just send it to you electronically.
I think the point of "they're fully backed" is the idea that someone who does have access to the banking system will give you 1 USD (or close to it) for 1 USDT even if Bitfinex can't. One could imagine a market where someone would give you $0.90 for 1 USDT and then that person would go to Bitfinex and get the actual USD that backed it (assuming it was actually fully backed). The idea being that Bitfinex might not be able to give you an ACH or wire transfer of your USD, but if you showed up at a bank in Country X with a suitcase, you could get cash.
I'd recommend the post from patio11 more than this comment. I guess the tl;dr is that it's not just heavier regulation, but that the US and many other governments don't allow anonymous money transfers. Banks and fintech need to know their customers so places like Bitfinex get cut off from the system. USDT is an attempt to offer people a way to turn something into a dollar-like thing without having access to the banking system.
