A variation of this question seems to arise on LR every few weeks. In general, many people greatly overestimate how fast the "average" person can run. You definitely need to define average. Is it all ages, M/F?
Broadly, a large percentage of "average" people have no hope of running a mile without walking, at any speed, let alone at 6:00 pace.
For a reference, here is some evidence from a corporate games mile race that my firm participated in a few years ago. We had 8 guys in our office run the mile on a track. All were aged between 25 and 40. None would be considered obese, but a few are on the heavier side (over 180). All of them work out regularly to some extent. Weights, elliptical, play basketball, crossfit.
4 of the 8 runners walked at some point in the mile race. And while you could potentially argue that maybe they went out "too fast," they all ran the first 400 in over 2 minutes. Out of the 8 runners, the "fastest" was around 8:15. I don't remember all the exact times but several were over 9:00. As far as I could tell, everyone ran as hard as they could. They just didn't have the stamina or speed to run anything approaching 6:00.
If they trained could they run better? Sure. When you run as slowly as these guys need improvement via training is inevitable. But these guys who are in decent shape by medical standards ran *2-3 minutes* slower than the pace you are talking about. The "average" person (even if you narrow it down to males 18-40) would struggle to run 1 lap at 6:00 pace, let alone 4 of them.