I have heard a theory that Jogo Do Pau would stem from sword exercises, either directly or possibly from people mimicking what they saw swordsmen do. I don't know if there is any proper validity to such a theory.
That the right hand is leading with a hand-and-a-half longsword comes natural, for several reasons including that it could be wielded with one hand only. But a thought that occured to me is that in armour, you often switched to a principle called half-swording, where the left hand grips the middle of the blade. This gives you more precision and strength in thrusting against armour. That means you are actually leading with your left, like in JdP.
So perhaps, this is how the left hand lead developed in JdP? Then you can parry in between your hands, as is done in a lot of other staff traditions as well. This also transfers to spearwork btw, and relates to how you manipulate rotational points. From what I understand, some styles of JdP, like in the Azores works even more with wide grips.
Of course, making simple grip changes and striking with a left hand lead could naturally evolve from there.
Just a thought...
Oh, and regarding the age of JdP, I could very well imagine it being well defined by the mid 1600, in other words around the time of Mairs short staff sections and the zweihänders. I remember that Luis Preto mentioned some JdP reference in a manuscript that I think was from about that time. Was it King Dom Duarte's book on jousting? From what I recall sword techniques that were similar to JdP were described there.
Oh, and this comes from Wikipedia:
"Some believe that it was influenced by an Indian dance or Indian martial art, which would have been imported and adapted in the period of the Discoveries, while there are others who say that its origins are medieval techniques of combat much similar to what is taught in the medieval book A ensinança de bem cavalgar em toda a sela ("The art of being a good horseman on any saddle") by Edward of Portugal (1391–1438). "
EDIT: I did a little research on Duarte and there is actually a few references here at ARMA to Portuguese wooden swords for training in Duarte.
"One of the clearest examples of the use of wooden swords in Medieval fencing comes to us from the 1434 writings of the Portuguese King Dom Duarte. In his Regimento, Duarte described: “And there he had weapons made of fine iron for other men, and he had lances and axes and wooden swords, and whenever he wanted to practice he armed himself with heavier weapons...made for learning different methods of defence and offence in which others are well versed.”[1] Duarte further advised: “Have spare weapons and armor at your house for anyone who comes over. Have wooden weapons to play with. When you spar use heavier weapons. Sparring helps you to learn new techniques from suitable partners. If no one comes over, train with anyone you can.”
http://www.thearma.org/essays/wasters.htm
It is even possible that the Portuguese knights taught lesser ranked and possibly poorly equipped infantry to use weapons like the goedendag with rotational strikes, but this is a very loose idea and I have no ground for such a claim...

It would seem logical though, that more skilled fighters would teach the less skilled, since it bettered the odds for everyone.
In relation to this, the above mentioned article mentions:
"From at least the tenth century in many Italian communities rival bands of youth battled in open fields with wooden swords and shields. (Davis. p. 14). In Medieval Pisa, there were also so-called giucco del mazzeascudo, or fighting “with wooden weapons sometimes altogether, sometimes two at a time; [with] a wooden helmet on the head and a padded breastplate of iron.” (R. Davis, p. 175, note

. Pretend wars between fighters wielding sticks and protected by shields, helmets, and body armour were a regular occurrence in Italian cities during the era. (R. Davis, p. 49).
John Stow’s 1598, Survey of London, recounted the following: “Ye may have read in mine annals how that in the year 1222 the citizens kept games of defence, and wrestlings, near unto the hospital of St. Giles in the field, where they challenged and had the mastery of the men in the suburbs and other commoners…The youths of this city also have used on holy days after Evening prayer, at their masters’ doors, to exercise their wasters and bucklers.” (Stow, p. 36). Starting in the late 12th century in Bergamo, Italy, an infantry training exercise using wooden weapons was called “battle with small shields”. In these pugne or “fights” all classes took part in the activity and Judges imposed heavy fines on anyone caught using iron weapons. By the 14th century it became a form of public entertainment often degenerating into brawls among the youth. (Nicolle, Italian Miltiaman, p. 31)."
This could also be a possible root for JdP...