In your effort to distinguish equestrian activities from cut-and-thrust and armored combat you neglected to mention mounted combat, which is virtually identical to armored combat in terms of weapons, armor, and rules, except that it is done while mounted on a horse.  The fact that two major equestrian activities are "martial art[s] where two combatants (or more, in melee) face off directly against one another" kind of throws a wrench into your argument.  

And that is to say nothing, of course, of the historical fact that one of the key distinguishing features of a knight was being a mounted combatant.  For obvious reasons of practicality that was not made a requirement of the Chivalry, but if one is to propose creating additional branches of the Chivalry, then surely equestrians who attain the appropriate level of prowess in mounted combat and jousting would be at least as good a fit as a cut-and-thrust branch.

Let me be clear: I'm not necessarily taking a position for or against the 'new branch' proposal in general, but it is difficult to argue that it makes more sense for cut-and-thrust than (combatant) equestrians.

-Edward de Kent
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