The only time of day I can play Destiny is after both my 2 year old and 2 month old are asleep, so talking over mic isn't really an option for me. It seems like by now a sort of etiquette for players who can't mic (or just don't want to) but still want to play the game to its fullest would have arisen. Something like, "If you want to join a fireteam for CoE, go to the reef and sit by variks. Or invite people you see already sitting by variks."

Same as always. Go to LFG, mention you can't use a mic and wait for an invite or just find people yourself.

Do such interactions already exist? If not, is there a good reason why not?

Social spaces have a small number of people, no text chat and the community on Reddit and such is pretty small next to the total fanbase.

let's make some! Using nothing but in-game gestures and social spaces, how would you broadcast intent to

Won't work for the aforementioned reasons.

(would raid even work? I hear they're much more involved and almost impossible without verbal communication)

It would work if you set it up before hand and you know what you're doing. Like if you're empowered on the right, tell them you're going to dance, so you do the dance emote. Someone will likely see it, know you're empowered and announce it. You might run into some issues, but nothing too bad. You could also consider just saying "empowered blank" just to avoid an issue. If you have a good team you would only need to say around six words the whole raid.