Should he have just coasted all the way down the hill? How fast would he be going by then? He seemed to do a pretty good job of braking as much as possible while still being able to steer enough.
Him going sideways meant his braking wasn't working. If there's slipping you want to let off the brake and down shift (you can do this in automatics) to the lowest gear that doesn't make you slip. Staying straight to where your wheels aren't slipping is a priority over braking or getting slower. The only way of slowing is to have traction, and you don't want it while sideways.
I would have tried keeping straight and then veered over into the dry lane and started slowing down.
Do you have the guy in back steering the rear wheels? That style of truck used to be commonplace when I was a kid, but I haven't seen one in a long time.
They're called tillers. The steering radius advancements and ladder construction allowed for shorter trucks that could turn tight enough to stop needing tillers in most areas. They're still in a lot of narrow and tight big cities like Kansas city, MO. The city I work at is also actually planning on getting one in about 5 more years, so they are still in demand. It's just that they aren't needed a lot of times, now.