Thank you Alexander! Seems like I needed a nudge here 
So this works now 
I added the main set of emojis, without variants (like skin and gender). That’s a set of 1869 icons (out of 3689 svg files in the Twemoji assets folder). I used tw- as a prefix, plus the english shortcode to name icons. One might still need to use a reference site to find the right emoji as the names don’t include any tags. E.g. this icon doesn’t show up if you’d type golf in the icon-picker:
I’ll share the component in theme-components when I’m ready with the theme. If anyone wants to try it out already, the repository is GitHub - nolosb/discourse-twemoji-icons: Use twemojis as icons in Discourse. · GitHub