I agree about the weather, it looks like the rain caused a down draft which translated into a strong wind blowing away from the rain, the same direction they took off. This is probably a common danger if you're taking off to beat weather.
I gave them the benefit of the doubt on the mixture, they said "mixture set" rather than rich, and it didn't look fully rich.
The flap setting is a very important point. I looked up a C172M POH online and definitely flaps 0 would be appropriate, particularly at high altitude.
I disagree about the speed though. Waiting for 65 knots to rotate or climbing at 65 knots would degrade performance. For a maximum performance takeoff the POH specifies
Flaps 0
Climb speed 68 mph (59 knots) until obstacles are cleared.
For a flaps 10 takeoff, 65 mph (56 knots) until obstacles are cleared.
Flaps 10 gives you a lot of extra drag. Too fast as well gives you extra drag squared.
(Obviously that is slow and could be a problem in gusty conditions like this. Gusty winds and a requirement for a maximum performance takeoff might be a good reason to stay on the ground.)
Often during training a few extra knots are added for "safety" and comfort. That doesn't usually matter because we usually have performance to spare. However, it can be deadly if you really need that performance.
If there is any doubt about takeoff performance, know what configuration gives you the best performance, and know what speeds to fly to get it.
However, whatever the reasons for the problem, the abort decision was good. Problems can happen to anyone, and it would be easy to sit there in denial and wait for the aircraft to climb until it is too late. Takeoff accidents are statistically much more deadly than landing accidents.