Hi Christie,
Yeah I've got the same, 2016 A4 B9 2.0 TDI. I had a garage do a flush, it didn't do much of anything at all interms of heat. However if you're planning on doing the heater core then I'd definitely get it flushed probably before doing so ( I stand to be corrected on this point ) that's just how I did mine and it's been fine ever since.
Interms of the matrix the actual matrix is fine, it seems a lot of garages qoute for this just incase the core doesn't sort the issue ( from what I can gather ). Did they run any diagnostic on the actuators to see if they can manually operate the interals flaps? Opening/closing? I only had one guy actually do this on diagnostic.
The core is between £60-£100 depending on brand, and it took me about 2hrs to do it. Was fairly simple... just a lot of screws in awkward places!