I've yet to have a need for an isolation transformer at home. As for safety, I guess the only 100% sure safe way is pay someone who is competent to do the work.
Normally,
I am the "someone who is competent to do the work".
For some 10 years, I spent much of my time working on Picture Monitors & at times Domestic TVs.
The standard SMPS in those days used BJTs, & from time to time these would fail, which sometimes took out other components.
One side of the circuitry prior to the "choke" (actually to all intents and purposes a transformer), so it was quite easy to get caught out using a normal CRO probe if you just run the DUT "bareback".
On the output side of the "choke", nothing was connected to either side of the Mains, so a normal CRO probe was quite safe to use.
We did, from time to time, have to look the early stages of the SMPS, which is where the isolation transformer came in.
You can get caught out, as I did one day, when someone else needed the isolation transformer, & I set to looking at the isolated parts of a SMPS then moving back to its input.
Powering the DUT down at the power point, I hung the probe between the test point & "common".
Turning the DUT on there was a "zap" & the bench RCD dropped out.
Damage amounted to a small burn on the ground clip!
Much of the troubleshooting with the SMPS could be done unpowered, using a Fluke 77, so it wasn't normally a major drama.
Most people who have been around for a while in the Electronics repair side of things have copped a "bite" of 120v or 230v & are still around, but such incidents have been a vanishingly small percentage of the time they have spent on such things.
It seems to be the trend to present working on power supplies as of the same degree of peril as crocodile wrestling,
but really, if people use a modicum of care, it is as safe as any other activity.
The other things I spent a lof time on were Radio & TV Broadcasting transmitters, which ran off 3 phase supplies, with HTs of around 10kv DC.