The design of the saxophone was not invented out of thin air but instead went through a series of evolutionary changes.

We'll start with the grandparent of the saxophone, the chalumeau, which caught on around 1630. It's built a lot like a recorder, but uses a single reed at the mouthpiece. Fingering is just a matter of putting your fingers over the holes.

As an aside...at its heart, every woodwind is just a resonant waveguide. The switch-up compared to a flute and recorder is that using a reed instead of an open fipple at the mouthpiece makes the instrument act like a tube that is closed at one end. The result is that the length to the first whole is a half-wavelength for a flute but a quarter wavelength for the reed instruments, making them much deeper and rich in the odd overtones.

As the 18th century began, the instruments began to be made with a register key which allows overblowing to a range an octave and a half higher. This major innovation created the first clarinets. In order bridge the gap between the upper and lower registers the instrument was given more holes and pinky levers to close those holes.

Adolphe Sax was already an accomplished clarinet maker when he went looking for an instrument that projected better. The crucial step was replacing the cylindrical bore of the clarinet with a conical shape that brought back the even harmonics and a more open timbre.

The very first saxophones were wood, but quickly changed to brass in the next iterations.

The cone shape of the saxophone also had the positive feature that it overblows just an octave higher, so now the player can learn the same fingerings for both octaves. A downside of the conical shape is that the holes have to get bigger as you move down the instrument and become way too large to cover with a finger; it really has to be a pad to close the hole.

From this point both clarinet and saxophone key hardware grew in complexity as makers looked for ways to simplify fingerings, improve intonation (i.e. in-tuneness) and allow faster motion.

As any recorder player can tell you, the way to get sharps and flats when there aren't half-step-holes is to use forked fingerings. For example, you might play a D by closing holes 1 and 2. You get a C if you close 1,2,3. If you want to play a D-flat, you close 1,2 and 4, leaving number 3 open. The chalumeau and early clarinets worked the same, but the note that you get isn't exactly a D-flat, plus you have fingers flapping all over the place trying to play these combinations.

In the 1850s, Clarinet makers fixed this by adding the "ring keys" which simplified the fingering patterns for sharps and flats. Flutes and saxophones did not have actual finger holes, but gained pad interaction mechanisms that were functionally equivalent.

To speed up parts playing in the pinky-key range, both clarinets and saxophones acquired many redundant keys where you could close the same hole with either pinky, so that if you had a passage that needed to alternate sides you could use either one. The layouts of both instruments are similar.

Finally to speed up trills, they both added keys just for that purpose, those keys that are played on the side of the right index finger.

TLDR: the current complexity of the saxophone was arrived through several decades of modification by many people. It started out simpler.

For more info, see Benade, Fundamentals of Musical Acoustics.