I can see that working very well!

Yeah, those buggers can be feisty. One thing to try is get a hold of it with channel locks, put a bit of pressure in the loosening direction, then give the handles of the channels a little tap with something; an actual hammer might be overkill, but anything to send a shock through the threads. Crescent wrench, screwdriver handle, lots of things can do the trick. We're basically breaking any dried schmoo in the threads.

Another piece to keep in mind is the part you're trying to turn is actually a bit flexible; if you squeeze too tight, you can pinch the outer threads against the inner threads, stopping it from turning. Adjusting the position of the pliers can make a big difference.

A comforting thing to know here is the faucet probably has several parts going through the sink; this means you can really put some sauce on to get it to break free without the faucet turning on you. Keep after it; it'll let go!

Edit words are hard