The most famous example is probably Gitlab (https://handbook.gitlab.com/handbook/company/culture/all-remote/asynchronous/)
Since their IPO the work environment seems to have deteriorated though (https://old.reddit.com/r/devops/comments/152o4bb/what_the_hell_is_going_on_at_gitlab/)
Curious to see if other people have any experience of real asynchronous work culture?
You’re conflating things.
Urgency is not a factor, but brevity and mental load. Writing mails takes time and forces you to jump between issues/contexts. Even if you answer all mails in one go, you still need to switch contexts again and again. Add to that the inevitable misunderstanding in written communication and you end up with hours of work for simple questions.
Of course there are issues that can be resolved via mail/tickets just fine, but many can’t. Forcing employees to choose a certain channel is not a good idea.
Why are companies going async? To go global. Now find a slot that works for Central/Western Europe and California. Good luck.