Just watching a road rescue programme on telly. I would not do what you do for all the money in the world. Changing an offside wheel on the hard shoulder in the dark, pissing with rain, freezing cold. Suicide. My heart is in my mouth just watching.
Surely, for your own safety, you should try to hook it up and off the motorway before changing the wheel.
That was our policy, unless there was a bloody good reason not to. That usually meant that the car was too low to load on the truck without a lot of time-consuming extra work.
I rarely had more than a quick look at the fault on a hard shoulder, and that was to decide if it was worth stopping somewhere safer and trying to repair. But you rarely fix modern cars at the side of the road.
You missed it being the 15th job of the day, 20 hours into the shift.
I don't miss it
There is a good case for not changing your own wheel, but if you
know it's going to go smoothly(no locking wheel nuts, proper wheel brace, adequately inflated spare, working jack - things we've all checked recently right?), then I would suggest that doing it is safer than waiting.