So, it's true that screen readers have heuristics for detecting layout tables. But those heuristics aren't perfect. They're not even that sophisticated, at least the ones that I worked on. And some (most?) screen readers automatically announce any table that they don't detect as a layout table, even when continuously reading straight through the document. Notably, the screen reader I developed myself, which was the first one I routinely used, doesn't automatically announce tables unless the user is manually moving through the page, because I wanted to minimize verbal clutter when casually browsing the web. But when I went to work at Microsoft and started using Narrator to read my work email, I found the tables in certain HTML emails very annoying. Granted, Narrator didn't yet have a layout table heuristic at that time, but even when it did, that didn't entirely solve the problem. Luckily, there was also an overhaul in Narrator's verbosity levels while I was there, and that allowed me to eliminate the clutter of layout tables by reducing the verbosity level in my personal Narrator settings, perhaps at the cost of missing some other things.

Nested divs per se aren't a problem. Implementing a widget as a div without the appropriate ARIA markup is a problem.