Very trueAFAIK, in OTL the Electors of Saxony managed to strengthen their army to a degree making it a meaningful opponent to Prussia or Austria. Now, the part regarding the PLC and Russia is quite intriguing because the Russian policy regarding (and within) the PLC was seemingly mutually-contradictive. On one hand the PLC was considered a potentially useful in the case of conflict with the Ottomans (which seemingly would be an argument in favor of having it stronger) while on the other an idea was to exercise a maximum possible “influence” in it, which meant to keep it weak. As far as I can tell, after the WoPS these seemingly mutually-contradictive purposes merged into one: keep the PLC as weak as possible.
The PLC as a military factor after the WoPS had been looking upon with an extreme disdain and its potential usefulness was considered strictly along the lines of its territory being used as a supply base with an understanding that the Russian troops can march through it with an impunity and that any manifestation of the unhappiness would be crushed without a difficulty (*). Having few pet ...oops... pro-Russian magnates (pensions, some Russian state awards) pretty much guaranteed that the PLC would not be able to implement any anti-Russian policies.
Which means that in an unlikely case the Saxon rulers manage to improve the Electorate’s military capacity that extra capacity would not and could not be used to increase the royal power within the PLC. Both the Poles and the Russians would be against it.
(*) As did happen during the 7YW and the Ottoman wars of CII. AFAIK, the Polish resistance to the Russian usage of the PLC a territory during the Russo Ottoman War of 1787 - 92 (protests against Russian marching through it, selling supplies to the Ottomans, refusal to sell food to the Russian troops) was one of the underlying motivations which led to the 2nd Partition: in the case of the next Ottoman war there would be no need to go through the PLC lands. Of course, the Polish attempt to create a meaningful state with a reasonably strong army was another factor and an answer to your idea regarding the PLC as a meaningful Russian ally against the Ottomans (I’m not sure whom at that point the Poles hated more, probably the Russians; at least the Bar confederates had been raiding from the Ottoman territory triggering war of 1768-74).
Edit: One possible effect of the crippled Austria is that MT is not in a position to start annexation of the Polish territory by taking Szepes County in 1769–1770. Prussia alone probably would not be enough to blackmail CII into an agreement to the 1st Partition. OTOH, if the Hapsburg “empire” disintegrates as a result of the WoAS than there is no 7YW and the whole partitioning thing does not happen (ditto for many other events).
On your last point, the edit one...I could see a situation in which Austria might try to form an alliance with Poland-Lithuania as a means of countering Prussian and Russian power. It had always been Hapsburg policy since their earliest days to forge alliances and marriage contracts not just to increase their power but to defend their holdings from rivals. An independent Hungary would be less inclined to ally with Austria and would in any event likely fall back under Turkish influence if not outright control.