How to get a stuck Moen faucet cartridge out?
I have a Moen bathroom sink faucet Model  Number 4570.  It has been in place for about 25 years.  I am the original owner.  The cold water faucet finally failed and I ordered new parts from Moen (the cartridge plus various adapters and also their tool to aid in removing a cartridge nut.  After removing the faucet handle and the cartridge nut, I then tried to extract the cartridge using needle nosed pliers as per the Moen instructions.  After being in place for 25 years, it did not want to budge (not surprising).
I then looked on the internet to see if anyone else had experienced this problem and it looks like it has been a common problem.  I read all the suggestions and tried putting vinegar on it to no avail.  I was impressed with one proposal from an engineer in which he suggested attacking it from the underside by disconnecting the tubing and inserting a 5/8” rod and tapping it with a hammer.
I am 82 years old (also a retired engineer) and I did not feel like getting under the sink.  I then tried to attack it from the top.  I fashioned a loop out of some old pipe strap I had and I was able to attach the loop to the top of the cartridge with the screw that is used to attach the top extension to the cartridge.  I then fashioned a lever arrangement using a block of wood and the lever that goes on my floor jack.  See the attached photos.  With this arrangement, the cartridge just popped out with very little effort.
Here is a photo of the Pipe Strap loop that I attached to the top of the cartridge.  This photo shows the cartridge after extraction.

This photo shows the arrangement of the lever with the cartridge still in the housing.

I added some extra silicone to the two “O” rings on the new cartridge and then inserted the cartridge and finished the job.  The valve turns extremely easily – maybe I did not need the extra silicone.  Moen was very helpful in sending the parts (at no charge since I was the original owner).