Let's bring it back on topic..
Engineers like to use square-waves because it can show a lot of things (most of all FR issues) with just 1 or 2 different frequency square-waves with one quick glance at their oscilloscope within a few seconds.
Furthermore, it is also very easy to see changes when applying a (difficult) load directly at a single glance at a scope image. All the way from DC to MHz region with cheap gear.
So... very handy for an engineer. I use them as well for the above reason, a quick glance at certain aspects, but not surprisingly am an engineer without an armchair.
This is sooooo much faster than running sweeps (automated or not) when starting to testi/trouble shoot .
Does it tell you more than some full sweeps in FR (if the analyzer could reach that far), phase and distortion ... no.
That could be a reason not to use square-waves when one is going to run a full measurement suite anyway which can dive much deeper than a scope image.
That scope image is linear in amplitude yet our hearing is logarithmic. So small level signals aren't seen but small variations in the signal itself are easy to spot.
Rise and fall times of the square-wave (so zoomed in a LOT) can show specific problems at a single glance certainly with audio gear that has a lot of overall feedabck.
At least to engineers with enough knowledge/experience. Someone with little to no knowledge would have to be explained what to look for and why.
So yes, engineers do love square-waves and for good reason. For this reason engineer type people (regardless if they are considered armchair or not) would like to see a screen shot of a low and higher frequency square-wave... at least for amplifiers (so analog plane).
The question is are they essential and are they needed on top of a full FR/phase/distortion analysis under various loads where the FR is tested far enough ?
I have to agree with
@amirm here. They are not an essential measurement that must be there all of the important info is there in other measurements, or at least can be.
Square-waves are just
very handy for engineers that like to see several factors at a
quick glance.
No need to get tied in a knot over this and certainly not to dis other people, regardless who they are.