The article seems to just be complaining that anthropic arguments cannot explain the precise _value_ of the fine structure constant. Well, nor can Occam's Razor, but we don't seem to have a problem quoting that!
Well, Occam's Razor is just a heuristic, not a logical necessity.
Sometimes the more complex answer is actually the correct.
Occam's Razor is not an argument or explanation. It's a heuristic for discriminating among different arguments/explanations.
And anthropic arguments amount to "there isn't any explanation for the precise value of the fine structure constant!" Or "you should just give up trying to explain why it has the value it has!"
We wouldn't be debating it if the universe were substantially different, but that doesn't explain why the universe we observe is the way it is.
In other words, for some reason they expect the universe to be more "understandable" or "sensible" than this . . .
Of course the strong AP is ludicrous in the extreme ;)
There's no number system you can invent where the fundamental constants turn into simple integers or have obvious ratios.
So why do they have those values? What gave them those values? The anthropic principle is disliked because it rejects further inquiry and progress.
Unfortunately, that doesn't explain how these values are generated. That is the important question.
No, their opposition is to the fact that a physical constant is a mere accident, a mere facet coming with the universe as a set, and not explained by some more fundamental mechanism.
And anthropic arguments are like: "if they weren't apples, then they wouldn't be apples..."