Why does a stock exchange need a clearing firm?

I see on wikipedia that "DTCC settled the vast majority of securities transactions in the United States and close to $1.7 quadrillion."

OK, so it seems that DTCC is the clearing house that everyone cares about.

Then I read that the NYSE and Nasdaq have subsidiaries which are clearing houses.

As I see it often explained, Alice sells 100 shares of ABC @ $5 while Bob buys 100 shares of ABC @ $5. Each has their own broker. The brokers’ orders get matched on the exchange (say NYSE). Then, the exchange sends everything to DTCC for post-trade processing (clearing services).

Why would the NYSE use DTCC to clear their trades when they have their own subsidiary to do it? I can understand that from time to time, you might want to offload some work to a competitor, through "coopetition" but if it were the exception, then DTCC would not be settling the "vast majority" of transactions.

Then I thought it might be someone buys on Nasdaq while another sells on NYSE but that doesn’t make any sense as the exchange is there to match orders and, as I understand it, the same stock trading on two separate exchanges will only be "matched" by an arbitrager, not by a clearing house.

This link states "There are two major clearing houses in the United States: The New York Stock Exchange (NYSE) and the NASDAQ." I assume they are referring to the NYSE subsidiaries here.

So, if NYSE (or any exchange) has its own clearing house, why would it pay to use the services of DTCC for most of its trades?