If you don't want to be disturbed by notifications, there is the Do Not Disturb option to snooze notifications.
If you don't want to lose context, use Slack threads, which localize discussion around an issue to a single place.
If you have a fear of missing out because you're piping everything to Slack - have you considered segregating your alerts into specific channels and muting those you don't care about? (Alerts are a bad example, though - really, you should use Slack for a visibility record during an incident, but the only alerts you should care about should bother you with PagerDuty or OpsGenie - if they're important, they should bother you).
I do work for a larger company (some 400 employees) and we use Slack quite effectively. One thing that's really helped us is that, by consensus, we don't make decisions in Slack - everything of actual import should have a ticket to condense everything in. Slack becomes useful for us if we need immediate assistance, have questions or need to communicate in real time during an incident. It's also invaluable when you have remote employees (!).