I didn't stated that Peter was totally stupid person. I agree, that some decisions that he made proved to be quite useful to Russia. For me, both PIII and CII were quite average rulers in the terms of competence, difference between the two lies mainly in CII having better propagandists than Peter and CII eventually getting to rule in her own right, which created image of Peter as a weak, despicable person who was overthrown by his own wife. He should not have had listened to his uncle and arrest Catherine when he had a chance to do this. Having said that, I disagree about Czartoryski being no worse than Stanisław. Stanisław was a puppet, not treated seriously by anyone in Poland. He didn't even manage to gather genuine supporters from nobility, he had to rely on ridiculous group of foreign individuals with very bad reputation. Even "Great Sejm" and Constitution of 3rd May were not his ideas, Stanisław was cautious (but I'll give him that he had more realistic view than majority of envoys) and his vision of foreign and to some extent domestic policy was different than vision of so-called "patriotic fraction" (which was in the large numer fraction loving Prussian money). Czartoryski, on the other hand has his clients almost everywhere in Poland, is widely respected and is an heir to the 100 mln zlotys. Czartoryski is able to pursue much more indepedent policy than Stanisław, having money for bribes, 7 thousands of private army and his own clients, which aren't low qualified foreigners.
I didn't know about Rumianstev pro-Peter stance, and you're right in a further part of post. So the most likely candidate is Henry Hohenzollern, he wanted that crown and Fritz could support his candidature in hope of strenghtening his influence in Poland.
Selection of Poniatovski instead of Czartorysky was based upon the factors you listed taken with “minus”: CII (and by extension Fritz who approved of her choice) considered greater wealth, a big following and more brains as being the arguments
against C’s candidacy. Within very short time it became clear that she bet on the wrong horse: Stanislaw proved to be totally impotent on the issue of the “dissidents” rights (about with both C2 and Fritz did care) and she turned to Czartorysky for help. BTW, with all their wealth Cartoryskies started begg8ng for money. How Czartorysky as a king would fare is anybody’s guess but he hardly could do worse than Stanislaw.
PIII was not stupid but (a) he was indecisive and could not react fast to the unexpected situations (which his wife and the plotters knew quite well and counted upon) and (b) he was excessively assured in the legality of his rule and did not take measures to back it up with a military force.
Regarding Rumiantsev, he was loyal to Peter and CII did not quite trust him, at least initially, (*) but after the dust settled she could not ignore the fact that after the 7YW he emerged as the best Russian general and was genuinely liked by the troops (and his sister, Praskovia Bruce, was Catherine’s close friend). Still, during her 1st war with the Ottomans she initially appointed him a commander of the 2nd Russian army with a function to guard the border against Tatar raids and only after she became unhappy with commander of the 1st Army Rumiantsev was appointed its commander and won the famous battles of Larga and Kagul. After the war he remained governor-general of Malorossia (formally, a very important position but completely dependent from the central power; Potemkin held him short of money funneling all available funds to Novorossia). During the second Ottoman War he, again, was made a commander of a small undersupplied army while most of the troops and resources went to Potemkin (**) who wasted them on incompetently conducted siege of Ochakov.
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(*) If the rumors regarding his true father were true, even he had greater right to the throne than Catherine.
(**) The main reason for Potemkin getting his appointment was his wish to get Order of St.George 1st class, the only Russian high award that he did not have and the only one he could not get (by order’s statute) without being a successful commander of an army. Needless to say that he was clearly unqualified for this role by a character and (in)competence. Siege of Ochakov turned into a protracted affairs in which Russian troops suffered big losses due to the diseases, cold and malnutrition. When the siege was finally over, Rumiantsev (who was quite successful in Moldavia) had been removed from his position and replaced by Potemkin (the following Russian successes were predominantly due to Suvorov’s victories).