And you lose state, unless you've carefully designed your app around that problem by using something like Redux. Which is a fine thing to do, but not everyone wants to.
The chrome dev tools API has a neat function to inject new source code by only replacing all functions from the new code, thereby keeping any state. I've used this when building wright (https://github.com/porsager/wright) to allow for hot reloading of anything (no need for redux). CSS reload is also instantaneous with wright, so might even make sense for you to use with the setup you described - No need for copy pasting ;)
You can see it in action here https://porsager.com/wright/example.mov - The start is editing js code (a mithril app) that is going through rollup, but there's also editing of raw css files at the end.
Insanely cool demo. Learning Wright has been near the top of my TODO list for longer than I want to admit — I just never manage to get through the other items on my open source TODO list :(