Expanding a bit on @joojaa's comment, posting what I know now:
After Effects (AE) can do a lot more than what is possible with SVG+CSS, so it's not technically possible to just export an AE animation to SVG+CSS. If SVG+CSS is the desired end product, use another animation software tool than AE. (I'm not a designer, so don't know what to recommend.)
Exporting to a Bodymovin' JSON and running the animation with Lottie for Web makes it possible to do a lot more, but there are still some limitation, see https://github.com/airbnb/lottie-web#ae-feature-support. The result will be best if the designer knows about what works and what does not. I had issues with letter spacing being completely off and elements not being hidden when trying to use one of the optimized versions of lottie.js.