Growing up we had a hefer/steer butchered every year and stored the meat in the freezer in just the paper and never had a problem. In the last few years, we've had a steer butchered every 2-3 years. The meat has kept fine in the interim. We've tried putting meat in the fridge inside ziplocks while defrosting, and half the time the ziplock leaks, too. Put it on a plate or in a bowl in the fridge to defrost.
As for storing meat in the self defrosting freezers, don't do it for more than a couple of weeks. It won't just taste bad, it will be bad. Use a deep freeze.