The movement in that video looks reasonable to me, but you can't gauge ring and pinion gear backlash from wheel movement because transmission free play is involved. They could never be wrong enough that you would feel them slam or they would fail in short order.
The only way to check the gears, as was already mentioned, would be measuring the gears themselves using a dial caliper and marking compound and pattern the gears checking for run out. An experienced rear end shop can guestimate slop off the pinion yolk by dropping the drive shaft and feeling the gears that way, but I wouldn't rely on that alone.
If the gears are installed/patterned wrong you will know pretty quickly: they will become loud and they will wear out early, and any slop will become greatly exaggerated.
Be sure to properly break in new rear end gears to get maximum life. Usually the first hundred miles or so you shouldn't exceed 50 mph or drive for extended periods of time, allowing the gears to cool off. After the break in period, change the gear oil. It's always best to follow the instructions that came with the gears as each mfgr/vendor heat treats differently and recommends different break in procedures.