IRV actually has the opposite problem: "If I vote for my ideal candidate, am I impairing the good candidate's chance of winning?" For instance, if it were Nader/Gore/Bush with 26%/25%/49%, and (hypothetically) all Nader voters preferred Gore over Bush but not all Gore voters preferred Nader over Bush, then voting for Nader instead of dishonestly choosing the "lesser evil" Gore would end up electing Bush.
No voting method can entirely escape both of these problems. But newer methods like STAR voting balance them against each other and manage to minimize both.
Arrow's theorem applies only to ordinal voting methods. Approval is not actually subject to Arrow's theorem; it's not even well-defined whether it passes the criteria.
However, there are various voting impossibility theorems inspired by Arrow which do apply to approval. Most notably, Gibbard-Satterthwaite.
Sortition (that is, random selection) is a system for roughly-proportional representation; that is, for choosing multiple winners for a legislature so as to represent a population.
Approval voting is a way of picking a single winner.
They solve entirely separate problems.
(Sortition is nearly optimal if your only goal is to match the distribution of voters by a subset of equally-weighted representatives. Technically, it's possible to improve on it slightly by allowing voters to cluster themselves into a hierarchical structure of parties and factions, then performing sortition under the constraint that each party and/or faction must come within 1 seat of its correct proportional representation. This kind of technique is called "reduced-variance sampling" when used in sequential monte carlo sampling.
But sortition, or party-clustered sortition, pays no attention to candidate quality. Most people would agree that if there are two candidates that hold the same ideology, but the only difference is that one of the two has a medical problem that only allows them to work for 1 hour a week, the other one would make a better representative. Sortition cannot be fixed to take this kind of factor into account, which is a big part of why many people prefer more traditional voting methods.)
The "favorite betrayal" criterion is defined mathematically. In your scenario, as long as you approve Ideal, you are not betraying your favorite, even if you also approve Good. The only betrayal would be if you didn't approve Ideal. And in approval voting, there is no way others could be voting so that betrayal would be a good idea.
You are talking about a form of strategic voting — whether or not you approve Good. But it isn't favorite betrayal. In fact, it isn't betrayal at any level. All rational approval strategy (assuming a convex distribution over others' votes) is "semi-honest" in that you never approve a less-liked candidate but not a more-liked one. The only strategic choice is where to set your threshold, then honestly approve all candidates above it.
According to the Gibbard-Satterthwaite theorem, any voting method has some form of strategy, so in that sense the best you can possibly do is ensure that all strategy is semi-honest. Approval is optimal in this sense.
The article focuses on approval versus IRV, and the advantages it gives for approval are valid.
As you know, there are various other methods (Condorcet, STAR, 3-2-1, etc.) that keep most of approval's advantages, but sacrifice simplicity for better expressiveness. It would be as unreasonable for the advocates of such methods to try to undermine real-world campaigns for approval, as it would be for approval advocates to undermine those other methods.
Unintentional spoilage and intentional tampering are two different things. There's nothing disingenuous about mentioning one of them but not the other.
It's true that, depending on the implementation, approval voting can be subject to tampering. But the key point there is "depending on the implementation". There are any number of anti-tampering measures that are compatible with approval. Other comments here have suggested some of them (such as using "0" vs "+" as marks). Others might rely on technological fixes (such as immediate ballot scans by multiple independent non-networked devices) or social ones (multiple witnesses during all stages of ballot collection and counting).
It may not be possible to be 100% certain of preventing every possibility of ballot-counting fraud, but it seems that >99.9% is possible, and that's basically good enough. In well-administered elections, spoilage is a much bigger problem. Thus, it's reasonable to focus on spoilage.
As others have noted, proportional representation (PR) is the best solution to gerrymandering. (Not arguing that we shouldn't also support second-best solutions like nonpartisan redistricting and court challenges to the worst gerrymanders.)
The three PR methods in common use are STV-with-multimember-districts; mixed-member proportional (MMP); and open (or, yuck, closed) list systems. All of these (except closed lists) are decent, but have downsides. STV leads to very complex ballots; MMP leads to "two classes" of representatives; open list focuses your voting power on the partisan choice, but doesn't give you much power to help set the direction of your favorite party. And all three can lead to extreme party fragmentation and thus excess "kingmaker" power for splinter parties, unless there are rules against that.
It is, however, possible to design a method without any of these downsides. Perfection is impossible, but the Pareto frontier is, and none of the above methods are on it. Here's one that is:
No voting method can entirely escape both of these problems. But newer methods like STAR voting balance them against each other and manage to minimize both.