But can you tell? I cannot tell many of these when I hear them in the piano tuning context (I've tried and given up). But I can hear how much better the piano sounds after all that is done.
Then again, pianos are a bad compromise. There is no way to have octaves and fifths on the same piano, much less thirds (both major and minor). We learn to live with it, but it isn't right, for simple pieces you can tune a piano for that exact song and make it sound much better - but if you get much more complex than "mary had a little lamb" you run into a conflict and some key will need to be two different notes.
> We learn to live with it, but it isn't right, for simple pieces you can tune a piano for that exact song and make it sound much better
"Better" is always relative, so it's often more correct to say it is/sounds "different".
For example, a true temperament guitar can sound weird/"wrong" to some people. A lot of folks are used to those compromises just being part of the sound. Similar to a "honky tonk" piano: if you tuned it differently, it would come across as a timbre change more than simply "better" tuning.
In the grand scheme of things, there are professions and arts that were once considered essential to everyday life and in ways that we today don't even consider. The profession is lost to time, but so is the need. It's the in-between transitional state, where the profession is in the process of dying, that is the most painful period.
Some day, the last note played on a piano will be played and lament for the piano tuner will die with it.
In the case of stringed instruments, equal temperament isn't the only compromise; there is also stretched tuning, which seeks to match the harmonics (rather than the fundamentals) of harmonically related strings to one another (in a mechanical system where the harmonics are neither perfect integer ratios of the fundamental nor constant ratios across all of the strings).
Then again, pianos are a bad compromise. There is no way to have octaves and fifths on the same piano, much less thirds (both major and minor). We learn to live with it, but it isn't right, for simple pieces you can tune a piano for that exact song and make it sound much better - but if you get much more complex than "mary had a little lamb" you run into a conflict and some key will need to be two different notes.