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?
Imparting proficiency does not impart appropriate anatomy or physical ability. This can be inferred from a druid's wild shape, which says (emphases mine):
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. (PHB, p. 67)
If a druid who already has a weapon proficiency cannot use it in beast form, magically granting the proficiency to a natural beast will not give them the ability to use a weapon 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?
Nothing specifically prohibits it, so this is a DM decision.
Let us examine to what extent form restricts or permits 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. (MM, p. 6)
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, ettercaps with the garrote variant) 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, most humanoids do use weapons - but there are still 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. This is 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 tendon boots' or a 'spiked chanfron' as a weapon, I would think it reasonable that proficiency in them would be granted as part of "all martial weapons", and thus could be profitably gained by a greater steed pegasus or a rhinoceros.
A saber-toothed tiger might have "heavy helmet armor proficiency".
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 provide them proficiency in the already-existing weapon of nets, but rather then throwing them laterally, allow them to be dropped from directly above, for which the peryton's flyby ability would be useful. Speaking of dropping things from above, winged kobolds attack by dropping bags of rocks on opponents, and I would permit that to griffons and perytons as well.
Such fanciful uses of exotic weapons and armors would of course require extensive training of the steed - but it is just such training that the proficiency granted by Tenser's Transformation represents.
RAW, Tenser's Transformation should grant proficiency in already existing weapons and armor - I would argue that nothing explicitly prohibits steeds from using the weapons or armor which the steed's natural anatomy permits them to use (as defined by the DM). This would apply 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?