I appreciate that this is an article written to popularise and advance science, however, there are several questions I would like to raise, that, if taken at face value, amount to disinformation. For instance when you say that:
“Einstein's theories didn't say what that number was, but then he applied special relativity to the old equations of Maxwell and found that this conversion rate is exactly the speed of light.”
What exactly did the author mean by that? The fact that the author refers to, and which Einstein deduced, was that if light was a wave as held by Maxwell, Lorentz, Fizeau and most other Scientists of the time, then a person running abreast of a beam of light should see, the beam of light moving at a uniform speed. This is because waves travelling through a medium move at a constant speed. But if one were running in parallel with the beam of light, this is not what one would see, because it is an oscillating electromagnetic wave. So different parts of the wave would move at different speeds. If, however, the Lorentz transforms were used, the problem would be solved and all parts (almost all parts) of the light wave would be moving at a uniform speed. Although, anyone, even without the help of maths, can easily conjecture that this cannot be a complete solution.
One possibility, a very real possibility, I might add, especialy after Max Planck’s discovery of the discreteness of energy in the form of quanta, is that Maxwell’s theory is wrong! A plausible alternate theory using the aether can be found at:
https://www.academia.edu/37258409/The_Electromagnetic_Universe_docx
Remember it was the aether that everyone was looking for, the above theory gives a logical explanation.
Again, the fact that the permittivity of free space turns out to be exactly 1/137 (0.007) is just one of those mathematical co-incidences. I don’t think any scientist in his right mind would base any practical reasoning on the fact that the fine structure constant approximates anything.