I haven't checked the electrical supply stores, but if I opt to snake
the cable through the wall, my next concern would be the cable clamp
at the electrical box. A normal clamp fits in the hole and a nut gets
tightened on the backside. But, I'll only have access to the inside
of the electrical box. Do they make clamps that can be tightened from
one side only? I've never paid attention because I've never needed
one before.

There are plastic "snap-in" cable clamps that might help, but you'd
still need to get to the back of the boxes.

What's the normal procedure then for adding a cable to an existing
electrical panel? Surely there's a way to protect and secure the incoming
cable other than tearing into the wall just to tighten a nut on the
backside?

Anthony