There's enough grey areas and caveats in healing dynamics in FFXIV that there is no black or white answer to the relationships between healers, healer DPS, and healer performance.

Best healing practices are to ABCs (Always Be Casting) while doing mechanics properly. You don't necessarily have to be ABC'ing with only DPS spells though as sometimes a party is so inept you have no choice but to babysit every single party member, thus reducing the number of GCDs you can dedicate to DPS. With that being said, even in high healing situations where it consumes a lot MP, you still have access to Break for MP free damage during healing downtime

To go back answer your opening question - no healer DPS isn't mandatory. It never has been, it never will be. Your primary role in the party is to keep your party alive. Caveat (yes, there will always be a caveat) - Under min ilvl progression is an exception. Also, healer DPS could be the difference between a wipe and clear in an under performing PuG setting. I would consider this a "healer DPS is mandatory" scenario, but these are scenarios that players put themselves in to and not because of designer intent.

In my mind, what separates the healers who perform at a baseline level versus those who perform well are those who understand how to keep the party HP safe while maximizing GCD utilization. Again, this doesn't mean you need to be DPSing like a God of Destruction - it just means you're spending enough GCDs and oGCDs to keep the party alive through the content and mechanical mistakes while filling the gaps with DPS. I appreciate healers who take the time to fix player mistakes more than those who tunnel DPS. No one's perfect and people make mistakes - healers just need to rectify those mistakes.

...yes even those annoying players who don't seem to learn from their mistakes after the tenth time. But that's another topic of discussion! (please don't be one of those players, lol)