This question gets asked (and answered) here many, many times. The general category of answers are as follows: people that HAVE used their TDI cars to tow, have done so repeatedly, and successfully, and have never had an issue. The people that have NOT, are frightened by what they do not understand.
Consider this: the A4 platform cars are equipped,
from the factory, with mounting holes in the rear unibody rails for an "official" Volkswagen accessory trailer hitch that is sourced from typical vendors like Westfalia (who has been making hitches for Volkswagens longer than most of us have been alive), or Bosal or Oris. These hitches are rated for 1500 kg (that's a tad over 3300 pounds). The light fixtures in the rear have extra plugs in them... that go to the accessory trailer wiring. Now, I ask, why would Volkswagen equip their cars with mounting points for a 3300 pound hitch, and have spots to plug a trailer wire harness in (on later cars they even have an actual trailer control module that plugs in and mates up with the CAN bus) that requires no special tools or drilling, if these cars are absolutely "incapable" of towing anything? Are the laws of physics different in North America than in the rest of the world? And why would it be such a stretch for my 2800 pound Golf to tow a 3300 pound trailer when it seems perfectly fine for a 7500 pound F350 to pull a 15k pound trailer? Seems like the Golf would have a leg up on that math, wouldn't it?
And all you folks talking about DOT laws, have you ever actually been stopped? I certainly haven't. And I've pulled stuff clear halfway across the country and back on many, many occasions. I've towed other cars via tow dolly with my Golf. Repeatedly. Missouri, Illinois, Ohio, Indiana, Nebraska, Oklahoma, Texas, Pennsylvania, Virginia, Maryland, West Virginia, Kentucky, Tennessee, Alabama, Georgia, Florida, Michigan, and probably other states I've forgotten. Never once an issue. I passed a highway patrolman going through the mountains of West Virginia in my Golf going UP hill with a '92 Jetta in tow!
And our little "Pudge" camper is happily drug behind it, too!