As a commenter, I find the dogpiling notifications are the most intrusive part of such a situation. When a situation like this occurs to me I usually quickly disable notifications on the comment in question and move on with my life and it really makes it much more tolerable. I may go back to check on that comment manually from time to time to see if anyone has managed to come up to anything new worth replying to, but typically after a day or two that seems to become pretty unlikely. The comments themselves don't bother me much, even if they are repetitive. I don't mind that people feel the need to have their own personal voice heard on a particular topic but I don't personally need to keep having it hammered into my brain every 10 minutes with a new notification either, I think it's the notifications that turn it into badgering.
I think a lot of the toxicity might be addressed by allowing users to set a limit (with a reasonable default, even) of the number of times they'll be notified about responses to an individual comment. Ideally you would also be able to enable/disable this on a per-comment basis, maybe you do want to be notified of every response to a particular comment because you're basically polling people for their opinions, and of course you would want to keep the existing functionality to disable all notifications related to a particular comment because like I said I use that all the time. But maybe we could limit it to a default like "up to 10", and then disable notifications, because after 10 replies, you probably aren't getting much new information. It would make the storm of replies when something gets particularly controversial or viral or popular a lot more manageable.
Of course if you get into a reply back-and-forth with someone, and you're arguing with a bunch of different people at once, that's not dogpiling that's on you.