The secret is to not be interrupted. If you're already reading the text, you might as well go and answer.
If you want to change something you will need to stop being interrupted (close the IM window, put your phone on silent, ...) and check once you have time. If you don't have time for a while, possibly give it a quick skim in case something important happened.
Things will always slip through the cracks. If you attend every interruption, it will once in a while interrupt an interruption itself and you're at status quo. At worst, put things on a todo list.
It's possible to do it.
It would work like this:
A list of firewall rules about who/what was able to send notifications. The firewall would be able to bump up or down the priority of a message, or discard them. Including rules being able to match say a regex inside the text, sender, time of day filtering.
And then at some specified interval (I would use 25 mins), I get all my notifications that didn't meet the emergency criteria in the firewall.
The iOS Focus mode does seem promising though.
As someone that lives life like this, yes, that's the point. If it's really important, they can ping me again and remind me to respond. Or when I have some downtime I'll peruse through my messages and emails again and stumble upon it and remember to reply.
(Sometimes I genuinely missed something interesting but this is rare.)
I would very much like to do what you say but with a million interrupts a day, it is now or never. If I neglect / defer something now, I would likely get back to it next week. Even for work. I do find time to focus 4 hours on some work activities but those are just the high priority visible stuff otherwise maybe if someone did not remind and make it a priority maybe it was anyways not a priority. But then things slip through the crack once in a while.