I’m not understanding the directions.
They seem to indicate that you’re supposed to remove this locking nut
or at least that’s what the second picture with the pliers shows, but then the third picture seems to still show the lock nut in place.
If that’s the right nut to remove, yes, I’d try penetrating oil on it. My preference is PB Blaster, but there are many others. Get one and hose it down, a few times. Let the oil soak. Tap it with a hammer from time to time, that can help get the oil in around the corrosion.
Vicegrip pliers may provide enough grab to turn the nut. Maybe. Regular slip joint, needle nose, or whatever those are in the picture, aren’t going to work. Maybe on a clean new unit at the factory, but not now. If there’s room, I’d probably just put a small pipe wrench on it.
Good luck pulling the old cartridge out with pliers. Again, probably worked ok at the factory on new parts, but you’ll likely need a lot more oomph than you’re going to get pulling on a pair of pliers. Moen sells a cartilage puller for their shower valves, something like that may work for this as well.
The last shower cartilage I replaced came out in tiny pieces, and took all night. I started in the evening, and went out for breakfast when it was done. Not what I had planned on, was supposed to be a simple remove old and install new.