Of course they could operate. They would just have to backdoor their encryption. Which, presumably, is what this legislation wants to achieve.
They don't want a world with no chat apps, they want a world with chat apps they can listen to.
What Signal is saying in this blog post is that they would rather give up the US market than weaken their encryption. Which is worth saying, because it's probably not true for most other apps. Most corporations would not give up the US market, no matter what compromises they have to make.