We concluded that the approval-voting-based pope-choosing elections seemed to have been highly successful – nearly as successful as could have been hoped – vindicating the decision in 1294 by Saint/Pope Celestine V to adopt that voting system.
There are several ways this conclusion can be criticized.
Q1. How are the pope-elections to be described as "successful" when in many cases they elected some amazingly slimy corrupt devious Machievellian evil bastards? (Which, when it occurred, tended to stand out in especial contrast versus the supposedly pure and godly religious preaching of the Church.) And when, clearly, some of the elections were "bought" by massive bribery?
A1. There are several answers to this. First, the sliminess and evil of some of the popes has to be judged in comparison to all the other rulers (emperors, kings, dukes...) also present in Europe at the same time. In my view, the popes were mostly comparable to those others, plus in some respects (i.e. helping encourage art and education) better. The Church catalysed some enormous evils such as the 9 crusades and (what may have been an unexpected consequence of them) the black plague. These things might not have happened on nearly as large a scale if Europe had remained fragmented without the Church as a unifying influence. On the other hand, it also is possible that, without the Church, something comparable or even worse might have happened, or that without any unifying force, Europe's harmful fragmentation would have lasted even longer. Later, the Church played an evil role in attempting to suppress science – but it could be argued that its positive contribution to learning outweighed its negative one during the approval-voting era; Galileo's persecution by Urban VIII (who also took nepotism to new heights by squandering an incredible 105,000,000 scudi of papal money on enriching his family) only occurred after the rule-change abolishing approval voting.
Certainly the reader of this history should take home the lesson that powerful theocracies are worse for people than secular democracies. But that is not the fault of the voting system used within that theocracy.
Second, the "success" of an election method has to be measured by how well it delivers what the voters want. In our case, the voters were "Cardinals." The Cardinals evidently did not care terribly much about today's notions of "evil" or "corruption" and seem to have welcomed bribe money. (Some of their subjects may have cared about that – but they had no vote!) What the cardinals mainly cared about was feathering their own nests, making sure they did not get killed or demoted, and maintaining the Catholic Church as a powerful entity for a very long time. Did that work? For the most part, "yes." Indeed, the papacy is unrivaled as the longest-lasting elected position on the face of the Earth and between about 1050-1600 AD was one of the most powerful; and the Cardinals were mainly very rich and powerful people throughout that era. In other words, by that definition, it was an enormous "success."
To bring this contradiction home in the most dramatic possible manner: a reader who feels that the Church was overall a Bad Thing For Humanity, presumably would have considered it best if the Church had used a very poor voting system, since that would have made it collapse as quickly as possible!
You can't judge the church's voting system based on its effect on humanity; you must judge it based on its ability to get the voters what they wanted.
Q2. How do we know that other voting methods would have done worse?
A2. It seems likely to me that either Borda, plain plurality, or instant runoff voting (IRV) would have done worse.
Instant runoff tends to favor extremists while the systems the Church used favored "nonthreatening centrist compromise candidates." I suspect that IRV would have given too much power to too many extremists and thus ripped the Church apart and caused frequent religion-based civil wars (believe me, some Popes were already crazy enough, extremer ones definitely were not needed); or turned it into yet another trying-to-be-permanent (but often not) family dynasty headed by the Orsinis or some such family. (Several rich powerful rival families seemed perpetually to be trying to turn the papacy into exactly such a dynasty, but the voting system almost always automatically acted against that.) You can look through the election stories yourself trying to judge when that might have happened – for me the answer is "enough." Either way, that would have been the end of the Church as a monolithic all-powerful unifying and comparatively peaceful entity.
With plain plurality, considerably worse winners would have been delivered on average. Also both it and IRV might have catalysed "two-party domination" and a consequent polarization and fracture (and certainly comparative sterility) of the Church. As it was, there were often identifiable rival factions but no long-lasting "two parties."
The enormous vulnerability of Borda to strategic voting (which basically turns it into "random chaos") would have been a total disaster given the huge strategicness of the voters in these elections.
Even with the system used, the Church still suffered the 39-year "great schism," and later (starting toward the end of the approval-voting era) the rise of the Lutherans, both of which constituted heavy damage. But the schism still would have occurred with essentially any voting system, since it was a product of essentially unanimous votes. And it appears that the rise of the Lutherans was insufficiently comprehended by the vast bulk of the voters until it was too late – i.e. was not the fault of the voting system but rather the voters – and anyhow on 8 November 1620 (only 1 year before the end of the approval-voting era) an international Catholic army massively subsidized by the Pope crushed the forces of the Protestant Elector Palatine, Frederick of Bohemia, in the Battle of White Mountain near Prague. At that point it appeared to all that the papal Counter-Reformation was triumphant and Protestantism finished. I.e. it appeared that the Lutherans had had their fling but would now be reabsorbed and crushed as usual. This perception later turned out to be incorrect, but only after the approval-voting era had ended.
Q3. What about all those bribes and strategic-manipulations?
A3. Any voting system would have been placed under enormous pressure by that. It seems to me that the system they used, resisted it comparatively well. It usually bowed but did not break. When it did break practically any system would also have broken, and a successful counterforce usually came somewhat later.
Q4. Isn't all that just a subjective judgment by you? What if I have a different subjective judgment?
Well... yes. There is a lot of subjectivity in the reading of history, as compared to (say) a "randomized clinical trial of some drug." Sorry, I cannot escape that. All I can do is advocate that you read about it and draw sensible conclusions. There are a few unchallengable objective key facts floating around though, such as the unrivaled lifespan and influence of the papacy and the power and wealth of cardinals. It is also worth nothing that Renaissance Venice and Ancient Sparta also had extremely long and successful histories based on range voting.