I think that he goes astray in a few places. First of all, that quote from Beethoven. He says "It appears in William Kinderman’s 1814 essay on Beethoven’s piano music in the Cambridge Companion to Beethoven" which makes it sound as if Bill Kinderman wrote an essay in 1814. Not so! Bill is a musicologist working today, and probably the leading authority on Beethoven. He and I used to teach together at the University of Victoria. I also think that Tom Service's interpretation of the quote goes astray. Beethoven is likely talking about the new style of piano playing that was coming into favor around then, exemplified by figures like Hummel and Dussek. Far more about finger-virtuosity than the likely more musically profound improvisations of someone like Mozart (whom Beethoven may have had in mind) or Beethoven himself, for that matter.
The other big error is in what he says about equal temperament: "Beethoven would have heard Andsnes’s instrument as being crazily out of tune." No, he wouldn't. The big shift in temperament from Pythagorean or meantone to equal temperament, that we use today, took place in the 17th and first half of the 18th centuries. By the time of Bach, it had largely been completed. Between meantone and equal temperament were a variety of compromises, many of them the work of a theorist named Andreas Werckmeister who published some works in the 1680s and 90s in which he argued for a kind of well-tempered tuning that would do away with most of the unpleasantness of the meantone system. Bach's Well-Tempered Clavier is an obvious reference to Werckmeister. This work, often cited as being the first example of equal temperament, was probably not intended to be played in our modern system, but in one of the ones worked out by Werckmeister. But it was close to our modern equal-tempered tuning because all of the keys were usable.
Beethoven, of course, coming well after this, and using all the keys with equal freedom, wrote for equal temperament. Oddly, one thing that Tom mentions completely disproves his case: "the distance between the C minor of the first movement and E major of the slow movement is a shocking shift from one world of feeling to another, which he then makes a joke of at the start of the third movement, translating G sharp to A flat." G sharp and A flat are the same note only in the equal tempered system. Tom is completely confusing the distance between C minor (three flats) and E major (four sharps) in the circle of fifths with the difference between G sharp and A flat in Pythagorean tuning: it is called the Pythagorean comma and it is why the use of remote keys like A flat is only possible in equal temperament or something close to it.
It is bad enough when writers in the mass media--or even in program notes--try to completely avoid any technical discussion of music theory, but it is even worse when they get it wrong.
Now let's listen to Beethoven's Piano Concerto No. 2 in B flat major, op. 19, which was actually composed before the No. 1. This is Steven Lubin, fortepiano, with the Academy of Ancient Music directed by Christopher Hogwood. Brace yourself, because the instrument is one from Beethoven's time so it sounds quite different from a modern piano.