AFAIK, Firefox does, if you're running the beta/nightly channel (and maybe you need to have set an about:config pref too, I forgot, search about it). I install the beta release of uBlock Origin from its GitHub releases page.
EDIT: browser makers are understandably cautious about it, even with a closed ecosystem it's already hard enough for them to avoid half their users getting pushed into installing dubious/malware addons within three hours of browsing, and their resources are limited. So, they default to closed-ness "for the masses", but there's a knob. I don't find this unreasonable.
It'd be neat if extension stores could integrate with some CI system(s). The store could just tell the user "built from <source URL>". Best of both worlds.
EDIT: browser makers are understandably cautious about it, even with a closed ecosystem it's already hard enough for them to avoid half their users getting pushed into installing dubious/malware addons within three hours of browsing, and their resources are limited. So, they default to closed-ness "for the masses", but there's a knob. I don't find this unreasonable.