I actually just bought 4 new tires for my car yesterday, I have always replaced 4 tires at a time, regardless if they were absolutely necesary all the way around or not. If you are replacing with the same tire/size, just keep the old (good) tires as spares if you have the space to do so.
However if you ARE replacing 2 tires, I agree that they should be put on the rear, though that leads to a catch 22. Keeping the better tires in the rear and more worn tires up front will lead to you replacing 2 tires at a time each time that your fronts wear out (since the fronts wear faster than the rears, and the fronts are already more worn down). If you put the better tires up front then in most cases a "typical" driver will have a greater chances of spinning out, but you are more likely to [eventually] have all your tires at the same tread depth since the front usually wear faster, at least in FWD cars.
Long story short, I think it largely depends on the time of year and driving skill. If you are in the winter/rainy season, better tires in rear. If it is dry/summer, throw the better tires up front so that you can get yourself towards a better cycle of replacing 4 tires at a time. Once you get into that 4x tires replacement cycle, make it a point to rotate the tires every couple oil changes. In a FWD/RWD vehicle I would definitely put better in the rear for the winter months, however on 4wd I never bothered. I just kept doing my rotations and called it a day. I could correct oversteer with correct applications of throttle, and 4wd could keep me out of trouble if the conditions got bad enough.
I know that last winter I had better tires in the rear simply because I neglected to rotate them. Worked great for me in my Altima. steering was fine, and as that video stated, I KNEW when I was getting to the limit of my traction and I could adjust speed/steering accordingly. I was never given a "false confidence" because my steering felt solid, only to find out that my rear end couldn't keep traction. Once summer rolled around I rotated the tires (better tires to the front) which made the car feel less stable in rain, but extended the time until I needed to replace the tires. This actually lead to one case where I was going around a highway on ramp in the rain (a bit fast, I will admit.. ironically I was doing so to test the wet traction) and my rear end let go. Suddenly I was staring at a guard rail. Thankfully I knew to point the front tires where I wanted to go and use the throttle to pull myself out of the slide...Most people in a slide like that would have gone straight around or straight into the guard rail (would have used brakes, not throttle). I'm not trying to boast, but just pointing out that the video/new tires in rear theory makes sense and is the most stable way to go in most all situations.. If my fronts had the lower tread I would have felt them losing traction earlier not pushed it into the corner as hard, instead of pushing in hard and losing traction out back.
The only reason I actually needed to get tires now is because I had a front tire blow out on the highway, which left me with 1 blown tire, and 2 tires with low tread (maybe 5-10k miles left on them), so I just replaced all 4 since winter is coming and I would need to replace them within a few months anyway.