Other support chat tools have solved this for quite some time with a queue and display of estimated waiting time. You may not get an initial response immediately, but once an agent becomes available they're assigned the chat and do communicate synchronously.
In comparison, Slack has no way to indicate that an agent is already working with someone else in another channel, nor any good way to assign a specific owner--customer questions are dumped into a channel with X support agents as members, one of whom will hopefully respond.
In comparison, Slack has no way to indicate that an agent is already working with someone else in another channel, nor any good way to assign a specific owner--customer questions are dumped into a channel with X support agents as members, one of whom will hopefully respond.