(i've never replaced a shower screen but i've been fixing plumbing and resealing around my bath because my landlord is a landlord)
i would guess that the wall part is actually two parts and that the screws you removed will allow the inner to pull out, exposing some much larger screws which attach the outer to the wall. you might need to pry the inner out if it's become deformed or corroded
i don't think that the gray plastic pivot is broken - it's too symmetrical. the dents are probably guides to lock the screen in the open or closed position. but your new screen will come with fixtures and it probably won't fit what's there already. so the existing fixture will need to come off
you'll also need new silicone sealant to go around the new fixture and a sealant gun to apply it. and gloves. definitely do not forget to wear gloves because that shit will still be under your nails in a week https://www.wikihow.com/Apply-Silicone-Sealant