Can magic that makes the uncoordinated wizard with no weapon skills a martial juggernaut also give my griffon the basic ability to firmly grasp a short sword?
No. Imparting proficiency does not impart appropriate anatomy or physical ability. This can be inferred from magic (for example Polymorph and True Polymorph) that transforms a character's shape while they retain the same weapon proficiencies that Tenser's Transformation would otherwise impart. Just having the proficiency does not grant the ability to use it in the new form. Or, even more explicitly, a druid's wild shape, which says
You also retain all of your skill and saving throw Proficiencies, in addition to gaining those of the creature...You can't cast Spells, and your ability to speak or take any action that requires hands is limited to the capabilities of your beast form. (PHB67)
If someone who already has a weapon proficiency cannot use it in beast form, granting the proficiency to a beast will not give them the ability to use it either. However, the quote leaves open the possibility that some beast forms do indeed have the inherent capability to use these.
Are any of the mounts listed able to make use of any of the weapon proficiencies granted by Tenser’s transformation?
Possibly. Nothing specifically prohibits it, so this is a DM decision. Let us examine to what extent form restricts or allows weapons use.
A monster's type speaks to its fundamental nature...The game includes the following monster types, which have no rules of their own. (MM6)
Since monster type has no rules of its own, there are no rules which state that weapons use is prohibited (or required!) of certain types.
Griffons and Perytons are monstrosities. There are monstrosities that can use weapons (centaurs, doppelgangers) and ones that do not.
Pegasi are celestials. There are celestials that can use weapons (devas, planetars) and ones that do not.
Dire wolves, Rhinoceri, and Saber-toothed tigers are beasts. There are beasts that can use weapons (apes, giant apes) and ones that do not.
On the other hand, while most humanoids use weapons, there are those that do not (quaggoth, troglodytes).
The common denominator among all the different creatures that do use weapons appears to be the possession of hands, as the PHB quote about wild shape emphasizes. Not surprising, since the simple weapons and martial weapons already on the PHB weapons list were designed for animals that have hands. But could there be weapons that do not require the use of hands?
If the DM allowed "spiked hoof boots" as a weapon, I would think it reasonable that proficiency in them would be granted as part of "all martial weapons", and thus profitably gained by greater steed pegasi.
Similarly with "horn blades" on a rhinoceros,
or "heavy helmet armor proficiency" for a saber-toothed tiger,
Finally, griffons and perytons presumably make nests, a process which requires both strength and dexterity in talons and beaks. At the very least, as a DM, I would allow Tenser's Transformation to gain them proficiency in the already-existing weapon of nets, but rather then throwing them, allow them to be dropped from above (for which the peryton's flyby ability would be useful).
Thus, I would argue that since nothing explicitly prohibits steeds from using weapons or armor, Tensor's Transformation would grant proficiency in both already existing weapons and armor which the steed's natural anatomy allowed them to use (as defined by the DM), and even more so for weapons and armor designed to function for their own particular anatomies (as permitted by the DM).
Related: Can a druid in Ape wildshape use weapons?