Is there justification for everyone who wants to process payments needing to understand that type of legal document? Is there some major liability that people are agreeing to when they accept the T&Cs?

I honestly just don't know anything about card redemption agreements and google wasn't very helpful.

The agreements between payment services and merchants are some of the most one-sided legal contracts you are ever likely to encounter. Unless you are a huge merchant, they are probably non-negotiable. Typically they grant the payment service and/or its associates the authority to literally take money straight out of your bank account, impose often unspecified levels of fees for vaguely defined transgressions, etc. They will also typically make the merchant responsible for almost anything that goes wrong in terms of fraud, other than perhaps a few specific exceptions where facilities like 3-D Secure were used.

Stripe seems to be little better than anyone else with much of this, though as I understand it that is partly because the card networks impose their own terms and intermediary services like Stripe are essentially just passing them on without having much say about it themselves either.

To give credit where it's due, I've never personally encountered Stripe actually abusing those sorts of terms, and getting signed up with them in the first place was dramatically easier than the alternatives available at the time other than possibly entry-level PayPal (+ accompanying horror stories, depending on who you ask).