Although it will function safely (my opinion of course), the NM cable you have is not proper for a dryer.
Type SE cable also utilizes a bare neutral/ground but it for some reason is legal in existing circuits.
You have a 3 wire circuit existing. The black and white wires are hot (240 volts) and the bare was acting as the neutral/ground.
If you can replace the cable with a 10-3 with ground back to the panel and remove the bonding jumper in the dryer jbox, you will have a code compliant installation.
Customers house, I'd tell them they had to replace it.