How would one do that exactly? Like I said, I could just not respond right away, and then I either have to respond to a whole discussion later in a mess of threads, or ignore it, but either way I'll get labeled a bad communicator and hard to work with.
Don't talk to each team member one-on-one, that'll never work.
Talk to your manager, make it clear in terms they understand. That doesn't mean complain or whine to them as that will just push them away from the point that you're trying to make.
You need to REALLY make it clear to them not in your terms, but in their terms. Show them that every time they do this your productivity goes down. Show them that if it's happening to you, it's also happening to others too. Show them the research that says interruptions are bad. Show them the communication models that other companies are using that work.
Your manager isn't going to do the research for you, they're too busy managing others. The only thing that managers are trying to do is to reduce the complaining to a minimum. If that means the minimum is only you complaining, then they are going to optimise for only you complaining. If you actually show them the research, show them a list of rules, show them a plan that can be implemented, then you complaining is always going to be the minimum. You need to show them that if they follow these steps, then the complaining will essentially go down to zero, only then will they act on it.
With that attitude you will. Honestly though you need to find a communication pattern that works for you and your teammates. In most situations if you communicate your needs clearly you won't get much pushback. For instance closing your instant message and email except for periodic checkins and letting people know why might improve the situation.
You could start small and block off a certain chunk of time during which you won’t respond. And communicate that to people ahead of time and the reasoning for it. Also set your statutes msg during that period. It could catch on and more people could similarly block off certain periods.
I find that even where immediate responses are not required where I work now (and, honestly, everywhere I've ever worked), the Slack status works pretty well for when I'd like to have some "me time". I usually set it to something business or "current fad" friendly, like "in my flow state - not monitoring slack or email".
How would one do that exactly? Like I said, I could just not respond right away, and then I either have to respond to a whole discussion later in a mess of threads, or ignore it, but either way I'll get labeled a bad communicator and hard to work with.