You do not use the debugger to run HTML/CSS. You can’t debug HTML and CSS. The debugger is meant for coding languages like JS where you set breakpoints and step through the code execution. But I never really used the VS Code debugger, if I want to debug some code I just add a debugger statement in the code and do it in the browser. In-IDE debugging is just for convenience.
For normal static site development with a bit of JS code, the Live Server extension is perfect. As for any performance issue with Live Server, I’d check out the fork I gave a link to.