The title asks it all.
The W.H.O., and principally because using Greek letters seemed to be a way to avoid prejudice and stigma that derives from ad hoc naming - which had often named variants after the geographical locations where they were first identified.
https://www.nature.com/articles/d41586-021-01508-8
As for the apparent skipping around - I think it’s just that only certain named variants have turned out to be significant.
Nu and Xi were skipped on purpose, one because it’s too easily confused with “new” and the other because that’s the name of the Chinese premier.
Who names the Covid variants…
Yes, that’s right 
The TAG-VE (or whatever they are calling themselves this week) only names certain “variants of interest”, to be specific. It was supposed to be some sort of mass PR thing for, how did they put it, “non-scientific audiences”.
ETA why Greek letters? This was discussed in an earlier thread. Basically, they went with that after rejecting proposals like “One, Two, Three, …” and Greek gods.
Anyway, at least we can now place the blame for all this COVID business squarely where it belongs - with the Greeks.
finally, i know who to hate.
Not because it’s the name of the Chinese president, but rather because it’s an incredibly common surname.
It’s a rather uncommon surname, in fact. There are several Xi homophones, none are in the top 100 surnames, and the common surnames are held by a much greater proportion of the Chinese population than the common surnames in Western cultures.
List of common Chinese surnames - Wikipedia
But with a billion people, of course even an uncommon surname still means a lot of people.
I also point out that people don’t agree on how to say “xi”. Sure, other letters may have variations across the pond (e.g. beeta versus bayta), but only “xi” seems to be said differently by everyone. I’ve heard “sigh,” “zigh,” “ksai” “ksee.” And that’s without getting into how other the last name Xi is pronounced.
And Jenkins might not be an extremely common Anglophone surname, but if WHO announced they were now tracking “the Jenkins variant”, it’d probably make some people pretty upset.
The upside is that we’re more than half the way through the Greek alphabet so there can’t be many more Covid variants.
This. Take it from a Karen.