I don't think Async solves the problem you mention -- that is more of a cultural thing. We expect folks to message each-other frequently (preferably in a public room with an @ mention) but set the expectation that you don't need to respond immediately. The public aspect is important as you will often get "drive-by help" from someone else on the team!
I have phone set up to not disturb me with notifications between 10pm and 7am so I wouldn't mind as long as nobody expects a response before next working day morning. Before I could do that I simply did not have company's chat/email on my phone.
My day starts with reviewing any received messages (no matter what channel -- email, chat...) and responding to them. I would find it unprofessional if someone did not. If this wasn't just an occasional oversight, then I would take it up with them.
Yeah...this is why I close down slack when I don't want disruptions. There's no point in a DND if others can do it anyway. It shouldn't be an option. If someone at work has an emergency and needs to get hold of me, they can give me a phone call. Non-verbal communications have a tendency to be abused, but the reality of actually having to have a conversation with the person you're disturbing means only serious calls seem to get made.
It's a much more scalable approach than trying to read the backscroll in Slack, IMO.
It has a few special features, but on the whole it's basically like a company-wide journal.