Yes, you are liable for ANY damage to the vehicle. I frequently rent vehicles from AVIS throughout Germany (most of my rentals are there) and you will be billed for any tire, glass or body damage. AVIS DE has been the most strict with me when returning the cars. They even check underneath. I was once charged 400€ for scrapes under the front bumper.

350€ seems a bit much for tire damage, but what kind of vehicle was it on? My rentals in Germany have always been a BMW, Audi, Mercedes or Jaguar and I would hate to do any 'real' damage to the vehicle. For what it's worth now, I'd recommend everyone to purchase their insurance coverage. Yes, you still get a "bill" for damage upon return, but the insurance policy covers the damage and the bill is never charged to your account or credit card.

I would wait to hear back from AVIS Customer Service and you may be able to plead your case to them. There was one time a couple years back where I had 3 rentals out in Germany and one of the cars didn't have coverage by mistake. A wheel had minor paint damage (curb rash) and I was billed 125€. Customer service was able to reverse the charge just by calling and explaining the error.