A little more info, still no code to share. Sorry.
CuteEditor is still intruding itself into the mix of circumstances that cause the unresponsive script behavior. I'll explain how.
On my three-CuteEditor page, I have found that removing my CustomValidator controls allows the page to function properly. I don't understand why. Now the mystery deepens, as all good mysteries do.
I have another page that is very similar to my three-CE page, except it has only a single instance of CE. (Both the three-CE and the single-CE pages have input forms with custom validators.) The single-CE page functions fine, even with the CustomValidator, which cuts against some of my earlier discoveries. I thought maybe the problem was having multiple custom validators. On the three-CE page, I removed all but one of the custom validators, but no dice, the unresponsive script recurred. Next, I went back to my single-CE page, and added a second instance of CE, but no additional validators. The unresponsive script recurred. This suggests the necessary ingredients for the failure to be at least one Custom Validator, plus multiple instances of CE (or at least my user control that contains an embedded CE).
So CE has not been totally eliminated as my culprit, tho' it is certainly not my prime suspect.
I do notice many references to __doPostBack in the files cuteeditor_gecko.js.aspx, CuteEditor_Gecko.js.initialize.js, Cuteeditor_safari.js.aspx, and CuteEditor_Safari13.js.aspx. But I don't understand enough of what that code is doing to hazard a guess.
I hope I can condense all this down into a small sample page that repros the problem. I am not at liberty to share the whole thing, unfortunately.