Dealing with frozen ground beef
There is a common misconception that you should absolutely never cook meat from frozen or near-frozen. This is incorrect. I would also not recommended putting any meat on a low heat to thaw it out - you are asking for tough meat at best and food poisoning at worst.
The aim when cooking meat is to bring the internal temperature up to a safe level for a sufficient amount of time that any pathogens are destroyed. When you're talking about a large roasting joint, then it is absolutely a good idea to fully thaw it (in the refrigerator), because it takes a relatively long time for the indirect heat of an oven to get to the centre of the meat.
However, with smaller cuts of meat you intend to cook with direct heat, cooking from frozen or near-frozen conditions is absolutely fine, provided you take extra care to ensure it is properly cooked. For example, there is no problem with slicing a chicken breast when it is still partially frozen (in fact it's easier to slice) and sauteeing it straight away. In fact, you can buy frozen, pre-sliced chicken that you fry straight from solid frozen.
With your ground beef, provided the centre was not so rock solid that you couldn't break it apart, you would have had no problem simply frying it as normal, though it may well have taken a little longer to get up to a sizzle. If it was rock solid, you could speed up the thaw by wrapping in plastic wrap and running it under cool water for a while.
Furthermore, there is virtually no benefit in bringing meat to room temperature. Even for small cuts like steak, the amount of time needed to bring it anywhere near room temperature will put it well in the zone where you should think about throwing it out, according to the USDA.
Serious Eats tested this:

I pulled a single 15-ounce New York strip steak out of the refrigerator, cut it in half, placed half back in the fridge, and the other half on a ceramic plate on the counter. The steak started at 38°F and the ambient air in my kitchen was at 70°F. I then took temperature readings of its core every ten minutes.
After the first 20 minutes—the time that many chefs and books will recommend you let a steak rest at room temperature—the center of the steak had risen to a whopping 39.8°F. Not even a full two degrees. So I let it go longer. 30 minutes. 50 minutes. 1 hour and 20 minutes. After 1 hour and 50 minutes, the steak was up to 49.6°F in the center. Still colder than the cold water comes out of my tap in the summer, and only about 13% closer to its target temperature of a medium-rare 130°F than the steak in the fridge.

and in terms of the effect on cooking:

So when searing a steak, the vast majority of energy that goes into it is used to evaporate moisture from its surface layers. Next to that energy requirement, a 20, 30, or even 40 degree difference in the temperature of the surface of the meat is a piddling affair.