[Alex KL-1]

About the measurement... yes, this one requires the system functioning. Is good to measure "in a hurry", with the + lead of the multimeter attached to the point (negative from multimeter will be to circuit ground, like the channel selector shield or some easy point), and then powering up the TV, watching as fast you can the measured voltage (before damage). Just when you are able to read the stable measurement, turn off the TV.
Then you can repeat for the HOT PSU, to take note. Same fast principle.

Is interesting both measurements, tyo see if one or both deviates from normal, to try to find soem culprit.

Another possible test, and more challenging, to check the PSU, is to remove the HOT and uses a resistor to emulate the HOT current, to independent test the PSU, before HOT eating. Some old PSU reacts bad to no load, hence the need for some load: some linear PSU rises the output voltage without load, making no-load measurement useless; and some older SMPS explodes without load.
A small lamp for emulating load... it depends, some PSU can be damaged by the filament in-rush current.