“Did you check in the preferences of INDD the settings of the clipboard handling? Make sure “all info” is checked.”

SOLVED.

I had completely overlooked this setting and don’t know how it was checked to “text only” by default (even after a Preferences reset, as David suggested). I apologize for being salty, but I was watching one of my key workflow tools essentially get taken away, and Adobe’s help was basically ¯(?)_/¯. I would have had to crank out jobs with a ton more effort, when Clients were just getting used to the speed I was able to do them at. Thank you!

Found a workaround (sort of):

InDesign now treats a copy > paste as a character dump, but doesn’t preserve the fact that it’s in a table. Copying a table and opening the clipboard shows it’s being copied as a table correctly, but InDesign doesn’t like it being put back inline that way. The trick is now to select the table and go up to the menu and “Edit > Duplicate”. This makes a duplicate table, although it makes it directly after the table you’re trying to duplicate. Still not able to directly duplicate a table from page 3 to page 7, but it’s something at least.