Who? Adolphe Sax.

There is a nice slide show at http://www.selmer.fr/histliste.php?page=debut which includes the patent drawings that Sax submitted in 1846.

The invention was not quite the miracle that you are implying. The number of holes to cover is pretty similar to that of the clarinets that preceded it. Those had been added a few at a time over the decades to improve playability. Clarinets are available in brass as well as wood.

What really sets it apart from the clarinet is its conical bore which causes it to overblow at the octave rather than the third harmonic, and reduces the range a little. This shape also produces all of the harmonics rather than odd-only harmonics that give the clarinet its closed-in timbre.

The conical bore means that the holes are too big to be covered by fingers so you have the comparatively huge pads as you get near the bell of the instrument.

For more reading see Fundamentals of Musical Acoustics by A.H. Benade.