I'm failing to see how that's worse than FPTP, which in practice would also result in picking Theocracy A (because the 5% representing the Secular Tolerance Party would be one of those "third parties" which get blamed for spoilers under FPTP).
Well yeah, in this particular instance they do. That doesn't mean they always do. For example, with the commonly-used Tennessee capital election scenario (https://en.wikipedia.org/wiki/Instant-runoff_voting#Tennesse...), IRV would select Knoxville while FPTP would select Memphis - still not quite optimal for everyone (Nashville would probably be the ideal winner in a "perfect" electoral system), but good enough for the majority and a strict improvement over FPTP.
Contrived scenario for FPTP picking the Condorcet winner while IRV fails to:
Back to the "which theocracy, or do we just tolerance?" question, but now each religion has at least two candidates and secular tolerance has a bigger constituency.
19% S, ...
16% A1, A2, S, ...
15% A2, A1, S, ...
14% B1, B2, S, ...
13% B2, B1, S, ...
12% C1, C2, S, ...
11% C2, C1, S, ...
I think that works out, FPTP makes the right choice for sort of the wrong reason while IRV makes the wrong choice. But I do think the much bigger concern is FPTP-with-strategic-voting behaving better (or at least no worse) than IRV-with-whatever-happens-in-practice.
Yes, there are games which can be played that increase the chances of some minority group in the second round, for example not listing secularism in the hope that it won't make it to the runoff. But it's a dangerous game of chicken for which the optimal strategy becomes apparent only after the results are tabulated. Alaska is a good example where even with excelent information and pooling data the Palin voters were stil unable to coordinate to skew the results. The existence of that risk is heaven and earth compared to an FPTP election which pretty much guarantees theocracy in your example.
Either I failed to communicate the situation I intended or one of us is very wrong about how IRV works.
Reducing the society to 20 people to simplify the proper language a little: We have 7 ballots that start "A;S;...", 6 ballots that start "B;S;...", 6 ballots that start "C;S;...", and one ballot that starts with "S". IRV considers the options in first place, sees that S is the least (only one person ranked it first), and eliminates that option. Which theocracy is picked will depend on things further down the list, but it will never be secularism. In the best case it will pick A, which would lose to S 13:7 in a head to head vote.
If we believe that "the appropriate compromise" between people with genuine disagreement will typically not be anyone's first choice, IRV specifically systematically excludes that appropriate compromise.
Yes, FPTP doesn't fare better here mechanically. The advantage of FPTP (which, to be clear, I still think is terrible) is that most voters understand that they need to vote strategically and understand how to do that. IRV is often sold on removing the need to vote strategically, and it does not do that, while making the correct strategic choice more complicated than "the better of the front runners".
The IRV case here is simply a cheaper way to organize a delayed runoff; voters are usually adept at understanding the implications of strategic voting, as long as it's framed correctly: in case your preferred option does not make it to the second round, would you still go to vote, and which would be your other choice? In Europe there is a long tradition for the two round delayed runoff, but it's certainly not presented to the voters as "you can chose anybody in the order of preference and it will be magically solved by the system".
Two round is in almost any case superior to FPTP, so if you can get it by simply tweaking the ballot paper and allowing second and third choices (IRV), it's an easy upgrade.
Approval voting might be closer to a Condorcet winner in some cases, but it's not a panacea to center squeeze because it's vulnerable to honest voting. What we see in practice is a tendency to function as FPTP, as all sides have an incentive to demonize the center and train their voters to mark a single option. So, in your example, the Sunni or Shia Muslims which can't stand themselves enough to approve each other but would still prefer islam in the second round (if secularism is not available), would lose the approval vote to a well disciplined contigent of Buddhists.
IRV solves that by not providing any help to competitors in the first round, and moving the "lesser evil" decision into the second round, encouraging more honest voting. Short of some Condorcet method (which will never be practical or intelligible by the average voter), IRV is an "acceptable compromise".
Getting a little more concrete:
Imagine a society split roughly evenly between 3 religions. Let's say 35% follow religion A, 30% follow religion B, 30% follow C, the remaining 5% belong to other religions or no religion. Approximately everyone with a religion (in our model; reality is more complicated but not sufficiently to make things inapplicable) prefers that society be organized as a theocracy according to their religion, would sufficiently accept secular tolerance, and violently opposes living under another religion's theocracy.
In this imaginary society, what is "the desire of the electorate"?
If everyone votes their honest maximal preference then the plurality says Theocracy A, which is violently opposed by 65% of the population! Picking secular tolerance over Theocracy A is not just something I (probably) prefer from the outside, but is actually preferred by the vast majority of the population over what IRV would pick!