Yes you are right, they need IRCTC, in-fact they can’t do without IRCTC registered users, they require IRCTC user credentials at the end!
Ticket price marked as 100 by railway is quotes at 120 by IRCTC(assume 20% premium) and 132 by PayTM(assume 10% premium).
Payment partner charges a nominal 2%, I assume website with such large transactions would be charged even less.
To explain the technicals of ticket booking whole of the backend is managed by CRIS, BroadVision a NASDAQ listed entity used to do it earlier.
https://medium.com/@kirtika.singh29/irctcs-digital-transformation-a-theory-of-change-75944c29c354
CRIS was a part of Indian Railway, but later the railway board directed them to report to IRCTC