He points out that there's a simple enough pattern that Archimedes could have spotted and soon figured it out. That is true enough, but Archimedes could have got by, while being even less clever than that.
Let's consider a few simple facts:
(i) if a and b are close to the ratio √3:1 then 3b and a are also close to that ratio, with the error in the opposite direction and of similar magnitude
(ii) Hence (a+3b):(b+a) will be substantially closer* to √3:1 than a:b is
*(in fact it improves the error in the approximation of 3 by a2/b2 by a factor that rapidly goes to 2+√3, but Archimedes needn't have figured that fact out - he only needs to have noticed that 3b:a is also an approximation of √3:1, and (a+3b):(b+a) always seems to be a better approximation than he started with)
If we proceed in this fashion from the very ordinary starting point of 2:1 (and eliminate common factors as they pop up), we rapidly hit on Archimedes' ratio.
2 1 (a:b)
3 2 (3b:a)
5 3 (a+3b:b+a), which becomes our new a:b
7 4 (dividing out a common factor of 2 here)
26 15 (taking out another factor of 2)
97 56 (taking out a third factor of 2)
And you're there after maybe a couple of minutes of very simple computation, even if you're a much worse computer (in the original sense) than Archimedes doubtless was.
Not in the least mysterious, and a bonehead like me figured it out almost immediately. I'll wager a master like Archimedes realized something like this in even less time than it took me. It's so simple, he probably wouldn't have even bothered to write it down.
What I have above appears to be basically the Bablyonian Method, on which Brent at The Math Less Traveled writes here.
I must have seen mention of this before (at least the phrase "Babylonian Method" sounds somewhat familiar). I have no doubt Archimedes was familiar with some version of it.