so what this guy is doing is dangerous?
Yep. Every metal part of his scope is potentially live, and perhaps other things as a result. If he were to use the isolation transformer on the DUT (television in this case) that would not be the case.
I certainly wouldn't do it.
Back when BJTs were "king" in SMPS, I repaired more of them than I like to remember, & if I absolutely
had to probe the thing for waveforms, I floated the DUT, not the 'scope.
You might have noticed in the video that most of the actual fault finding was done using a DMM on the Continuity or Ohms ranges, with the 'scope just being used to check the correct pulses appeared, so he could have just "buttoned things up" & seen whether the TV worked.
I found that after fixing a few SMPS, most of the time, I did my testing without the need to apply power.
I have seen 'scope floating done, but this was in a specific situation where testing needed to be phase to phase on a 3ph system.
The 'scope was placed on a dry, wooden "Technician's stool" to avoid inadvertent earthing.
To do the actual tests a primary team of three was involved: -
One to observe, another to stand by & yell loudly in case of trouble, & yet another standing with his hand on the main breaker handle ready to open the circuit if he heard a yell.
There were several others who were doing other work nearby, but "keeping a weather eye" out for trouble.
Measuring 50Hz phase to phase "hands free" in such a situation is relatively easy, as the 'scope can be set up to "line trigger", & the volts/div set for a convenient range in advance.
The "three people" setup was rigidly adhered to, however, & unless you can reproduce such conditions, "floating" a 'scope is a
very bad idea!

To bring things into context----'scopes with isolated inputs were rare & specialised in those days, & differential probes cost as much as the original oscilloscope.