For patting dry meat, what is a substitute for paper towels?
There are only four ways that I know of other than towels (paper or otherwise) to dry meat:


Air circulation
Time (in a relatively dry environment).
Heat
Momentum


Most people avoid the heat approach, as you'll start to cook it once it's hot enough to be safe for long-term storage of meat.  Some recipes may start in a low oven to dry the surface, then remove it, let the oven pre-heat to a higher temperature, then finish cooking.  (as it's difficult to give recipes that know how quickly your oven heats up).

For momentum, you basically have to flick the meat such that the water gets flung off.  Which is prone to lots of problems (letting go of the meat, plus the spraying of contaminated liquid everywhere).  You could use a salad spinner, but if you did, I'd recommend keeping a separate one for meats, as you don't want to risk contaminating other ingredients that would be eaten raw.

For the airflow, you can set it under a low speed fan ... avoiding high speeds so you don't end up aerosolizing the moisture and flinging it through the kitchen. 

Or you can place the food in a ventilated container and leave it in your fridge overnight ... possibly with a battery powered fan in the fridge to improve airflow.

As all of these ideas have drawbacks (food safety, time, etc.), most people just accept the waste of using paper towels.  There are a few times when one of the others might be used (food dehydrating, trying to get a glaze to set up (eg, peking duck), dry brining, etc.), but they're relatively rare.