I use the vscode-restclient and my primary reason is the conversational flows you can build against an API. Does Hurl support this? If so I would absolutely switch. All I would need to complete the experience is a plugin to do highlighting and integration with the Hurl files.
But really Hurl looks really interesting, being editor agnostic is the best solution for your problem, I agree.
[1] https://github.com/Huachao/vscode-restclient