"The second of these quests, according to websites, "allows for the summoning of Majordomo" and hence the appearance of Ragnaros. "

Actually the part that allows for summoning is the last part of the chain.

"So, the question here is, do we need to access that quest so that when TG takes on MC on its own we actually have the option of fighting Ragnaros?
"
The answer to that question is yes.

"I would assume that only 1 person in 40 needs to be able to summon Majordomo, but if we wander in there and no one can summon him....."

Actually you need at least 7 people that have completed the quest to summon Majordomo.

"So, if no one has taken the time to do the Waterlords quests and gain faction with them, we might be setting ourselves up for failure. Anyone got any better information on this?"

You can actually complete about the first 3 or 4 parts of the chain before you have to have a Honored standing with the Waterlords. Your honor will also increase at a very good pace once you start killing mobs in the Molten Core. It is not uncommon to go from neutral to friendly in one or two runs just killing Luci and Magmadar. I would not waste the time farming for faction in Silithus. In all honesty most everyone will probably be well on their way to revered by the time you are ready to summon Majordomo.

After you have a Honored standing then the Duke gives you a quest to collect the 4 hands of the serpent bosses(Luci,Gehennas,Shaz and Sulfurion.) After the 4 hands are turned in then the Duke allows you to get the special water to summon Majordomo.

Hope this helped.