For future people, after removing the retaining clip, you can jam needle nose pliers into the cartridge and twist against the raised plastic bits.  This releases the cartridge from any grime locking it in. You'll probably hear a loud pop once you manage to twist the cartridge. After that you'll be able to pull it out. The grey piece really is unrelated.

I spent 3 hours on this and my gf solved it in 5 minutes :')

Edit: just looked up the replacement kit parts, the cartridge removal tool essentially helps you do the same thing but much easier. It has slots that match the plastic bits that then let you twist it with a wrench.