Definitely worth checking for the smoke. That's great, as any fault codes my help rule out potential causes.
I believe all VAG cars have an egr cooler with the egr. I've found this link, I appreciate it doesn't relate to your specific engine but it gives good info on egrs. On reading it, it could just be a gasket leaking. I do feel you could try rad seal but if it did work then irl would only be a temporary repair.
https://www.vwaudiforum.co.uk/forum/showthread.php/196553-2012-A6-quattro-3-0Tdi-EGR-coolant-leak-advice