Dryers are entirely 120V machines except for the heating element (for parts commonality with gas dryers). The neutral wire handles real live no-kidding working current - 100% of the dryer's draw on "fluff/no heat" setting is on the neutral. Most of us look at the 3rd wire as a lazy hanger-on that does no work (except during a ground fault)... but not neutral on a dryer! It's a no-kidding for-real working conductor on a dryer.
Why do we insulate neutral (generally) if it's near 0 volts when everything is working properly? Because if neutral gets loose (a commonplace occurrence that should not make anything dangerous) it energizes the neutral on the field side of the break. Worse, 3-wire dryers "ground" to neutral, so dryer chassis gets energized too and that's why 3-wire dryer runs were outlawed in 1996.
The 10/2 line as both the black and white wires hot. From what I can see that's "okay"
No. It was never OK. It was illegal the day it was installed, even if prior to 1996. Even prior to 1965 when grounding was not generally required.
The GROUND in 10/2+ground NM cable was never designed to be a neutral. It lacks the necessary insulation and thermal design, and in older NM it is undersized.
Prior to 1996, 3-wire connections were allowed, but only with cable with a certified neutral (but not a ground). Specifically
- "/3 no-ground" type NM/Romex cable, which largely went extinct after 1965 when everything but dryers and ranges needed grounding.
- Service Entrance cable type SE-U with a bare mesh neutral (and extra thick insulation to support that). Service entrances don't have ground because they run between the overhead utility supply wire and the meter, and ground isn't established yet. However utilities stopped offering services smaller than 60-100 amps, so 30 amp SEU has no reason to exist.
If you're thinking "Wow, when this rule was written in 1965, NEC deliberately allowed only cable types they knew were going extinct". EXACTLY! The plan was to sunset the 3-prong dryer connections as supplies of legal cable dried up. Once the only cable available was 10/3+ground, everyone would switch to 4-prong dryer outlets.
Of course, as those legal cable types dried up, some went "hey I'll just use /2+ground!" A lot of inspectors shrugged. This idea became a meme, but it's still a code violation.