My truing stand checks both sides of the wheel simultaneously so I don't have to flip it to get proper dish - one of the reasons I got it - saves me from having to get another tool.

Truing stands are unreliable for checking dish because they're subject to various errors.

Depending oh how they support the axle, they're vulnerable to errors introduced by a bent axle, or wear or distortion where the axle meets the stand. Often you can see these errors by positioning the wheel but not tightening it in place, then using a cone wrench to rotate the axle. Errors of the type I described will move the rim side to side, throwing the dish readings off.

Then, depending on the construction of the stand, there's always the risk of the centering calibration going off. The Park TS-2 stands are notorious for this problem.

However, there's no need to buy a dishing gauge, since a very accurate one can be made at zero cost, whenever needed, using a table, 3 soup cans, and a stack of coins.

I agree completely. But (as I'm sure you know) you can center the rim on a TS-2 or other any descent truing stand by reversing the wheel on the stand multiple times and adjusting the spokes until the rim is centered regardless of which way you mount the wheel to the stand. This can be done by using only one caliper arm whether you have one or two. With practice this can be done quickly.

Yes, but if you have a slightly bent axle (and don't know it) this is a recipe for going crazy as you try to adjust to a moving target.

Sure - but at that point wouldn't one want to replace the axle/whatever was damaged rather than trying to dish the wheel? I mean if you had to do it - you have to do it but using damaged components just seems like a disaster for any sort of precision when it comes to assembly.

I have a dish tool as well but generally only use it to check the dish after the wheel comes off the stand just as confirmation that all is well. If the stand goes out of center for dish - I'll simply build a wooden disc and put a solid axle through it and then use that to re-center everything. As with any calibrated tool - the calibration does need checked periodically and possibly adjusted.

Anything that can touch the rim and then be adjusted to reach the axle nut reliably will work :).

This should be apparent I would think on the truing stand.

The axle on my Trek is actually slightly bent [not sure how that happened] but I trued it up and checked dish and it was fine. It'll never be perfect but I doubt I could get it perfect with a bent axle even if I used a dishing tool religiously.

I was able to see the very slight bend in the axle as I rotated the wheel on the stand - this was likely the largest reason my wheel was out of true - I don't think that the spokes were really improperly tensioned but instead that the bent axle threw the whole wheel off giving me a wobble.

I was able to clean it up with the stand and get it to rideable condition but, that said, I'm looking at a couple of IGH hubs to build a wheel for this bike on :). 11-speed IGH sounds like fun :).

I just find it a pain to undo the wheel and reverse it. If I am going to get it out of the stand, I might as well use dishing tool which will give me a quite accurate idea of which way to adjust.

One thing that hasn't been mentioned is ensuring the bearings in the axle are in fact in good condition and properly adjusted. It's also a reason why I use the QR in the truing stand to tighten the bearings to their riding tension.

Yes, it also isn't subject to drift because of a loud party the night before or seasonal allergies, and can be used with different gage spokes with a quick reference chart.

The music tuning was an interesting example; firstly because a (quality) tuning fork is a standard unit with a known uncertainty, much like the class M2 and F1 calibration masses that I use to check my balances. On the other hand it's quite different from spoke adjustment because the frequencies involved with tuning a piano match with distinct harmonics.(that's the whole point, right)
Meanwhile every wheel out there has a different combination of spoke length, gage, lacing and rim properties which would create a different target frequency for each combination and little if any distinct harmonic, and a mechanic let alone a home mechanic generally puts tone recognition pretty low down the list on his resume (not because it's a bad thing but mechanics have a huge list of priorities that come before tone) unlike a specialist musical instrument tuner.
Though for factory production of hundreds of identical new wheels there is a company that uses a digital tone meter on it's building machine, I think it's Easton but their site is down now so I can't check.

Old methods work, new methods have been developed because the old methods have some weaknesses, pros & cons and all that.

If you use a gage, make sure you treat it well and give it some light oil like triflow on a regular basis.
I use to work in a shop where I would hand true about 3000 new machine built wheels per year, from various makes and models. Tension gages were preferred to fiddling with old timey methods for both consistency and speed. On a good factory wheel this would take about 3-5 minutes, on a bad one(with a slightly warped rim and crazy off tension) maybe 15.

The basic process when using a gage was: stress the spokes, check dish, test a few spokes in a row for tension to get a basic tension starting point, if way off target give every spoke a quarter, half, or full turn(depending on the starting tension and dish. Treat every spoke on one side with the exact same amount of turns or you may screw the radial hop. Do not stop half way around to check tension, answer the phone, fight off a bee, or anything else.) note any spokes that feel like outliers double check them with the gage and load balance as necessary,
Stress again, check a few more spokes,
When tension is close, move on to truing, and true with balanced turns to avoid throwing off the radial or the tension. eg, if the spoke in the center of the wobble is on your right and needs to be about a half turn looser then take up the slack with 1/4-1/2 turn tighter on both adjacent spokes on your left. Small adjustments of about 1/4 total turns and less generally don't need balancing.

Stress again, check dish and true again, make fine adjustments, rinse&repeat as needed, ending with stress, check, done.

There was a direct correlation between turns and tension readings something like one full turn on each spoke would swing from 18-22 on the park gage, I don't recall the exact number as I haven't done it in about 2 years, it was not quite linear though, more turns were needed to adjust one point at very low tension than were needed at very high tension.