I'm not being hyperbolic here. Number 1 and 2 are how people quickly use vague citations to Arrow's Theorem to shut down talk about voting reform even when the status quo consists of provably terrible systems like plurality voting.
Number 3 is the true result that if you relax the rather overly-strongly defined IIA criteria with a much more-reasonable criteria -- that the winner must remain among the top rock-paper-scissors loop of the voters -- then Arrow's theorem simply doesn't apply. This is well known: that "top loop" is the Smith Set and every Condorcet method of voting satisfies it.
There's also another interesting result: if voters have merely "single-peaked preferences", such as opinions about where to set a volume knob, then Arrow's theorem also doesn't apply since there will be no rock-paper-scissors set of equally fair options.
His theorem says nothing about the way things should be, but rather about how things are. I mentioned Poundstone's Gaming the Vote in another post, but that book in particular discusses the implications of the theorem and analyzes better alternatives to current systems.
For example, you confidently dismiss plurality voting, but you can't really do that unless you establish what the goals of your voting system are. These goals need to be clearly defined, you can't just say your goal is "making sure everyone has a voice" or "making sure the best candidate wins." That's really all Arrow's theorem says: there are several criteria that most of us agree are desirable, but you cannot have all of them.
Arrow calls scissors an "irrelevant alternative" to the race between paper and rock (since scissors doesn't win but makes paper lose). This is just unreasonable -- if there exists a rock-paper-scissors loop among the top preferences of voters, then the voting system should at least acknowledge that reality and make sure it picks someone within that loop (the particular winner is somewhat arbitrary).
And, indeed, if you instead use a more reasonable criteria that requires just that -- that the winner must come from the smith set -- then all of the "reasonable" criteria are satisfiable. Hence the overhyping of Arrow's theorem.
You are right, though, that basing analysis of voting systems on rigid criteria can lead you into trouble. Over-reliance on Arrow's theorem is just an example of that.
IIA is an extremely reasonable quality of an individual's preferences. Although you could probably construct pathological cases where it might sort of make sense for an individual to swap his preference of X and Y when Z is introduced, I contend that it would be exceedingly rare. Showing that IIA is impossible to guarantee (at the same time as other criteria) in an aggregate of preferences is a pretty big deal.
Arrow's theorem does not say or logically imply that, if there is a cyclical Smith set, the voting system shouldn't acknowledge that and pick someone within that loop.
The article alludes to one very important corollary, though IMHO it doesn't explain it very well: Arrow's theroem is like the CAP theorem - it seems to be a much stronger restriction than it is. In other words, if you're willing to make just a few very straightforward assumptions (ie, compromises), you can create a system that appears to "satisfy... rationality of their preferences".
Nobel laureate Amartya Sen has demonstrated that if you assume that there are certain rankings of preferences that are rare enough to be ignored altogether, then instant-runoff voting will in fact satisfy all the constraints of the Impossibility Theorem.
Let's use the 2000 US Presidential election as an example. There were three main candidates in Florida (Bush, Gore, Nader), for a total of 6 rankings. While Nader played a spoiler role, Nader and Gore shared more of a platform than Nader and Bush did. So it is very reasonable to assume that there are few people who would have voted for Nader over Bush, but Bush over Gore. This reasoning allows us to eliminate a number of those 6 rankings - and more importantly, enough rankings that the criteria of the Impossibility Theorem are likely to hold.
 The article does mention Sen, and alludes to this finding, but I don't think it explains it very clearly.
accepting dictatorship in the former or non-availability in the later gives you consistency. Reminds about USSR - high consistency (at it least it was initially) with very limited availability of salami (all the way until the last day of USSR).
The trick there is the "apparently quite reasonable" part. Far too many people take it as a given that those assumptions are sacred, which leads to a form of worship about this theorem. The IIAC in particular is problematic, and if that is relaxed, it's not quite so certain anymore that a "perfect" vote-counting method is impossible.
Similarly, if an election has a Condorcet Winner, and adding a candidate creates a Smith Set (thus potentially violating IIA if a certain candidate is chosen in a "tiebreaker"), this is not necessarily a bad thing.
If that happens it means that people always legitimately had those views, and were not able to properly express their views by choosing among the more limited number of candidates.
I always thought an interesting vote gathering technique would be something that actually interviews/polls the voter. Ask them for their ordinal ranking. The voter would know that if a Condorcet Winner exists, then that candidate would be the winner, but also ask for various cardinal ranking numbers, along with their approval line. That way there would still be the incentive to vote sincerely, while the cardinal information could be used to break the Smith Set loops. (The only downside here is that some people claim that the presence of cardinal tiebreakers creates an incentive for people to vote insincerely to create a Smith Set...)
Alternatively, if a Smith Set occurs, schedule a second round of voting for only those candidates (like a Louisiana Runoff, but Condorcet style), so that voters could better educate themselves on the remaining candidates.
You can draw your own example of this if you like. Draw an equilateral triangle and its altitudes, creating 6 regions inside. Put some dots in the 6 regions in the middle (voters), but leave every other region blank. Now, declare the vertices to be candidates (a 3-way race). If you compare any two of them, and have voters vote for whomever they're closest to (based on which side of the altitude they fall on), you'll end up with a rock-paper-scissors situation.
Also interesting to me is that I believe the IIAC can actually uncover that kind of completely-legitimate cycle. Meaning, while introducing an additional candidate can never lead you from one Condorcet Winner to another, it can lead you from a Condorcet Winner to a Smith Set. If people change their preferences in that manner, then it means that they have found better choices for them. In other words, if IIAC happens, it could be an indication that the original set of candidates wasn't really appropriate for the voters in the first place.
Thinking about both at the same time is uncomfortable, because if Smith Sets aren't an indication of voter-population confusion that can be resolved with more education and communication, then it basically means that the more choice you offer, the less likely there will be one candidate deserving of victory. If that's true, then making the arbitrary choice (among most likely candidates) might actually be the best outcome. Not exactly democratic though.
What do you mean by "sincerely"? Can you demonstrate a situation where it is in a voter's interest to give a less preferred candidate a greater score than a more preferred candidate?
Approval gives a big advantage to those who can best predict how others will vote. Yuk.
There's no real way to simplify it beyond "here's a set containing the rank order of every voter" if you don't know in advance what election method will be used. In fact, you might argue that the rank order is insufficient, and you also need to know whether the voter would approve of each option in an approval vote, or 0-10 rankings for range voting, or whatever.
Condorcet cares most of all about people's preferences between pairs of options, so that's why you'd summarize it as a matrix made out of those pairwise preferences. Borda cares most of all how many times something is "first place", "second place", and so on, and cares nothing about the pairwise rankings; Borda advocates claim that pairwise rankings will just confuse the issue.
(Opinion: I happen to believe that Borda is garbage for anything where politics is involved. It might be okay for the kind of contests where it's in use, like the baseball MVP or Eurovision, except those are probably secretly political too.)
Given some objective, like "stay informed" or "make the best possible choice when I vote", and given certain problems like "news is corrupted by advertising" or "news is dumbed down", is it at all plausible to achieve any of those goals in 1 hour of news consumption each day?
A naive application of this theorem suggests not... "none, anyway, that satisfy certain apparently quite reasonable assumptions concerning the autonomy of the people and the rationality of their preferences."
In other words, might it be the case that there is little intrinsic value in consuming the news piecemeal, in the way that most of us do?
It just means that those "apparently quite reasonable assumptions" weren't reasonable after all and must be relaxed a bit according to a precise definition.
You just took a mathematical theorem and went into a completely unrelated tangent. It's as if someone says there's no point in keeping development schedule because temporal order is relative to the observer.
Please don't do that.
Keep reading. This one is more than just math.
To be cynical, in the context of voting, consume news however you want to because to some extent it doesn't matter.