4 managers to one dev is a ratio that warrants a very quick change of jobs.
No, I mean, some sort of cross-team functionality where you need broad agreement, not four managers all trying to manage literally the same thing.
I probably do more cross-team stuff than the average dev, but in anything beyond a trivially-sized organization, managers are doing it all the time.
I'd say the solution is to attack those problems directly.
There are some things that just work poorly asynchronously. I find heavy-duty explanations of something, where you need the highly-interactive back-and-forth is necessary, and honestly, even just trying to type something is wasting minutes vs. saying it, is a common use case for me. Getting 4 half-distracted managers who are lobbing emails at each other into a quick conference to resolve the matter is sometimes a really good idea, too, because the "lobbing emails at each other" is a political minefield. A lot of potential for hurt feelings and miscommunication that can be avoided just getting everyone in live voice chat, with their full attention, for even just 5 minutes sometimes.