Plumbing – How to get a wrench into this location to remove this nut

From the way you describe it, it sounds like it's a regular gate valve. You might be able to repair it as you describe by removing the packing nut and replacing whatever is broken or stripped out inside.

The only exception to that would be a ball type valve, but those will just have a handle that makes only a 1/4 turn. Those generally aren't repairable, but then again they almost never fail either.

Personally, I would just remove and replace the gate valve with a good ball valve. Gate valves have a tendency to rot out or seize up from lack of use. I actually had one once that disintegrated in my hand when I tried to turn it off (spraying water everywhere and making a nice mess). Ball valves are just much more reliable.

Don't bother with all that, just remove the U shaped pipe (The Trap) which should be a pressure fitting that can be easily unscrewed without any tools. Most of the crap will be in that U part of the pipe, the rest you can get out by sticking something like a toilet brush or pipe cleaner up the exposed pipe.

BTW: Be sure to put a bucket under the pipe before you unscrew it, some nasty water will likely leak out.

As for that pivot nut, if you really want to remove it, it USUALLY is easy enough to unscrew with your bare fingers, but if it is rusted you can use a pair of channel lock pliers. If it is just spinning however, it may be all the way off the threads and you just need to pull it back. It is attached to a rod that pivots in the middle of the pipe. You might need to shake it a little to get it loose, it will also probably help to detach the rod that goes from the sink stopper pull to that nut since it will get in the way.