Neutral Is Not Ground!
Seriously. They serve completely different functions, and should be isolated from each other entirely except in the one specific place they are bonded.
A lot of people have gotten into the bad habit of playing fast-and-loose with this. They wire main panels and jumble up neutrals and grounds on the same bus. They poach ground when they need neutral for a smart switch. Doing those things is arguable, but the line of thinking which leads you to them is wrong, dangerous, and has a body count.
Like ThreePhaseEel says, electricity wants to get back to source.
Neutral is the normal everyday current path for current to return to source.
Ground is a protective shield. It's used several ways, and the relevant one here is as a shield to protect humans from loose or misplaced "hot" wires. It should never, ever flow current normally. Any current moving on a ground is a fault condition.
Of course, current wants to get back to source, not to ground. Why does it flow down ground during a fault condition? Because as part of the design of this protection, we intentionally bond ground to source (neutral) inside the main service panel. Because of this bond, it is not wrong (merely sloppy) to have neutrals and grounds use the same bus in the main service panel. In ones way-of-thinking, they must be separated, and also in a sub-panel, they must be separated.
In the beginning there was no ground, only neutral. You'd wire a dryer hot-neutral-hot and use a NEMA 10 connector. However there were a lot of electrocutions, and they were searching for a way to slow those down. In particular, equipment was mostly all-metal back then, and a fault between hot and chassis resulted in the equipment being energized at 120V. So the idea of a safety shield came up, to connect all the chassis to what exactly? Something that would safely take energy back to source, so they bonded neutral to ground at the supply, the transformer- or fairly near it, the main panel. (Since the transformer was usually up on a pole, hard to inspect, and the ground jumper is hard to protect from damage).
They then wanted to roll out "grounding" to American homes. They did this gradually, as renovations were done. However, the dryer and stove industry pushed back politically - they didn't want their customers having to spend $1000 to replace wiring in order to buy a new stove - it would kill sales! So they politicked to allow NEMA 10 to continue in service, on the logic that the plugs are rarely unplugged and dryers are rarely moved. They said "just ground the chassis to the neutral". Errmmmm...
How that fails with dryers and stoves
The problem is, if the neutral+ground wire has any kind of breakage, the hot wires will "pull up" the dryer's side of the broken wire up toward 120V. If the dryer chassis is bonded to the dryer's side of the neutral, that means its chassis is now energized. Touching that with rubber soled shoes won't kill you, but touching that and anything else, like the washing machine next to it, or the oven... Will!
As you would expect, this situation has a body count. It tends to be among the poor, so it's under-reported.
Grounding that dryer properly
I'm not going to trust my safety on a political compromise.
Obviously, hot-neutral-hot need to be connected in the usual way for the dryer to function.
A huge safety win is to put the hot-neutral-hot onto a GFCI breaker. I would be confortable continuing a NEMA 10 in service in that situation. It would protect from any sort of current leakage.
Otherwise I would properly retrofit a ground, which is now legal to do. It can follow any viable route back to the same panel. Then use a NEMA 14 receptacle (which the dryer probably comes with) so you are paying to replace the receptacle instead of the dryer cord.
Or I would hack the dryer to be 240V-only and not use neutral at all, by internally adding a small transformer to power whatever it has that is 120V. (Probably the timer and controls, maybe a light bulb). This isn't that weird; this is exactly what the manufacturer does when they sell the same dryer in 240V/60Hz territories like the Phillipines. In fact I would just use OEM parts if available. At that point I'd use a NEMA 6 connector (ground, no neutral) and (illegally) re-task the old "neutral" to be a "ground" by marking it with green tape and moving it to the ground bus.
Or if I was poor, I would hack in a proper ground any which way I could, i.e. To a nearby water pipe in an all-metal pipe system... not legal but better than nothing.