Why 16, 33 1/3, 45 and 78 rpm?

Strictly 16 2/3 rather than 16, and was used for speech records where range didn't matter. The wider the range of pitch, the further apart the grooves (OK, I know it's only one groove really) have to be. Easy numbers for gearing, I suppose, in the case of the modern ones. The old ones had to be fast as the sound was mechanically produced and a lot of needle movement was needed to produce this. Electric reproduction enabled the 'microgroove' vinyl record to be brought in. Incidentally, 78 rpm wasn't universal. Columbia (UK, that is) and Regal turned at 80 rpm until the merger with HMV and standardisation. I've got quite a few of them. And I've got a very old one on the National label that appears to be labelled for 79 rpm! World records didn't have a fixed speed, as they changed speed depending on where the needle was. This gave a constant needle to surface speed (and a rather complex mechanical system as well!). They also ran from the inside to the outside, which gives an interesting chance for reverse messages if played on the wrong type of player. I don't know why 78 was standardised on, off the top of my head.

78.26 happened to be the speed a common, mass-produced 3600 RPM motor would yield if fitted with an equally common 46:1 gear.

Around this same time the original acoustic method of recording was made obsolete by a new, and far superior, electric system. The fidelity possible at 78 RPM could now be had at a much slower speed--33-1/3.

And then RCA came up with the 45 just to compete with Columbia's 33 1/3.

https://www.straightdope.com/21341404/why-are-vinyl-record-speeds-33-45-and-78-rpm-and-why-the-big-hole-in-45s

Technically speaking....More sound or VPS (vibrations per second) can be transmitted up the tone arm at higher speeds. This leads to more information for the ear to hear and the greater the sound reproduction.

Similar to the modern bit-rate of CD's and other electronic sound data.