Most likely the failure was in the diff-end CV cup, and didn't harm the shaft going into the diff itself. Even if you somehow split the whole cup, I suspect it would crack open and leave the shaft itself intact. Possibly mess with the diff seal surface.. but things shouldn't get into the diff.
A solid axle is a different story, since the part of the axle that breaks is usually within the same housing as the diff itself. Everything can mix up.
OP's strategy of leave the axle in place and remove the driveshaft is the best method for most of these issues.. it lets the broken CV only take on the role of keeping all it's parts in place. It no longer has to transmit any torque to the wheel, so you should be able to drive.. understanding it might make minor problems within the CV worse, but realistically they will likely just have to buy an axle anyway.
If the shaft itself had broken things get more complicated, since it would likely want to flop around everywhere. That would require removing the shaft and leaving both cups in place, or finding some way to sleeve it to keep everything close to center. Driving at high speeds with this problem would make me much more nervous.