Could anyone clarify the role of the clearing corporation (DTCC) vs. the role of Stock Exchanges?

I am a bit confused as to why DTCC is needed in the market. For instance, if I want to buy (as a retail investor) a stock listed under NYSE, does my broker ever have to deal directly with the NYSE or is everything performed by DTCC? How is it that the NYSE is not the entity that transfers entitlement from a seller to a buyer? What does the NYSE does exactly? Because until finding out about the existence of the clearing corporation, I assumed Stock Exchanges existed to do exactly what DTCC does.

I understand the need for ensuring that the trades are going to go through in terms of liquidity and the existence and location of the assets, etc, so traders can feel safe about transactions. My problem is I used to think this was performed by the Stock Exchanges, and I fail to see why a separate entity was needed.