Just to clarify a few things.
It is never a good idea to float a scope by using an isolation transformer or cheater plug. This can induce a lot of additional stress on the equipment. Also, if an electrical occurs failure within the scope, the chassis may be elevated to line potential. This introduces a serious shock hazard.
I see nothing wrong with using an isolation transformer to float the circuit under test. There is however, a right and a wrong why do do this. If you simply connect a standard isolation transformer to your mains powered equipment (DUT) while keeping the ground connection intact, you aren't really achieving any isolation at all. Using an isolation transformer in this way only serves to filter out EMI and other undesirable artifacts present on the mains. This is why they are used so extensively in hospitals to protect sensitive medical equipment from interference.
If you truly want to "isolate" the mains powered circuit you are analyzing, you need to remove the ground connection on the output of the isolation transformer while keeping the ground connection on your scope intact. At this point, if you measure from either end of the output winding to ground, you will see 0 VAC. The only way to receive a shock using this method is to simultaneously contact both output wires of the transformer. This is was you might call an isolation transformer for "technical purposes". The danger of smoking your scope by connecting the earth lead to an energized part of the equipment has been eliminated. The risk of shock has also been greatly reduced.
If you don't believe me take a few measurements with the ground connection intact and then lifted on your isolation transformer. With the ground intact, you will measure full line voltage from either end of the secondary winding to ground. Now lift the ground on the transformer. Measure from each end of the secondary winding to ground. That mains voltage that you just measured is no longer present. Probing either end of the winding to ground will not show any voltage because the winding is no longer referenced to earth ground. The only way to measure mains voltage in this scenario is to probe directly across the secondary winding. Presto, instant isolation.
OK,I'll bite!
If neither end of the transformer secondary is connected to Mains Earth,how do you "measure full Mains voltage from either end of the secondary winding to ground"?
This is what normal transformers do---they isolate the secondary from the primary.An Earth connection which is just basically a wire to the frame of the transformer,hence to the "Earth" pin of the 3 pin socket which carries the normal secondary voltage,has no connection whatever to the secondary itself,& cannot be at any voltage w.r.t that secondary.
Its only real purpose is to provide protection in the case of a primary winding-to-core breakdown of the transformer,so it really doesn't do anything for the DUT,in any case,whether connected or disconnected.
If,for some strange reason,it is connected to one side of the secondary,it is no longer an "isolation" transformer.
Even in this case,you will only see the same voltage as the Mains (not really the "Mains" voltage) between
one side of the secondary & Earth.
In the SMPS on which I have worked,the DUT chassis which is earthed via the 3 pin power lead is effectively isolated from both sides of the incoming Mains,as well as from the common of the high voltage rectifier & the switching devices.
If neither side of the incoming Mains is connected to Earth,the "earth clip" of the Oscilloscope may be connected to the "common" of the SMPS switching circuitry without risk.
PS: The gentleman in the video covered the whole thing very well!
"Isolation transformers" with one side of the secondary tied to Earth are not common in Australia---at least in my experience.
I've only ever seen one standalone transformer wired that way over many years,& that was used for another purpose altogether.
Your "full Mains voltage between Earth & either end of the secondary" must be a "phantom voltage" as referred to in the video.