Yes.
> Which is wrong
No. I even provided you with the reference to the paper in question. Did you even try to read it?
> as the fine-structure constant is one of fundamental constants of the universe and one _whose value is not predicted by any theory_
This is where you go wrong, and where you misunderstand Feynman's point.
The correct statement is that all respectable theories of physics (to date), including the Standard Model, include an irreducible number of values which must be plugged into them "by hand". In other words, once you've written down your theory, there are some parameter values in it about which the theory itself gives you no guidance; you could give them different values, and the theory would still work. It would just be describing a universe with different properties than ours. In order to make it describe our universe, you need to get those values from experiment.
Feynman's point is that we don't have a theory without at least some such parameters (even string theory, which he disliked, has the string tension, and it skirts the need for more by randomly picking a vacuum, which sets the values of low energy theory "constants"). It is not that the fine structure constant has some particular status as "more fundamental" than others.
The choice of constants which you can determine by experiment is constrained by the theory, but generally not locked down completely; you can choose your set of constants, as long as they are independent, i.e. as long as measured constant #1 can not be determined by plugging measured constant #2 into the theory and doing some calculation. If constant #1 can be computed given the theory and constant #2, then they are not independent, and the choice between them is arbitrary; a convention.
The choice between fine structure constant and magnetic moment of the electron is one such arbitrary choice. Given one of them and the Standard Model, you can compute the other. And It turns out that it's actually more convenient to do it this way: measure the magnetic moment, then compute the fine structure constant. There is no reason at all to regard the fine structure constant as more of an "input to the model" than the magnetic moment, as you claimed at the start of this thread.
Needless to say, Feynman knew all this perfectly well. You are just taking away the wrong message from an attempt to popularize the topic. "QED" was a popular book, not a graduate text.