Stocard does that for some cards, but not Woolies for some reason. I've added my (rarely used in these Kindle days) Dymocks loyalty card to Apple Wallet via Stocard, but the option isn't there for Woolies.
You're right there's no difference to the cardholder between MasterCard and Visa. Banks negotiate deals with Visa and MC, and most Westpac cards were MasterCard for years, although their new credit cards are Visa (debit cards are still MC).
Keep an eye on this list on the Apple site. As an aside, look at the crazy number of banks in the US!