I never formally learned Japanese but heard that in some songs. It's not いち、に、さん、し, and not just one, two, three, four, but mixed together. I didn't care at the first time, but after I heard that in another song I'm curious. I checked and found they were from the same composer/lyricist and singer combination, but there are also others according to Google. Is it just because some specific lyricist or singer likes this, and some other lyricists learn from them, or is it more common, for example also used outside musics? Even if it's the artist's preference, does it contain any kind of references, or just could be (re)invented by any normal Japanese person? And is there any specific reason that they are all switching to onyomi from 3 and stopping at 4?
As for popularity, I think it's fairly common — as I've heard this kind of mixed counting (including variations such as ワン、ツー、さん、はい "One, two, three, go!") — as a joke. I'm not going to explain why it's funny because it'll spoil the fun :) but at least I can point out that the loanword スリー takes three morae, which is overlong and sounds somewhat clumsy in the normal two-morae-per-number counting rhythm.
The reason why it stops at 4 is simply because it's a time-marking shout in order to synchronize rhythm before the next musical bar. Most songs have quadruple time, so we count four beats.
I don't know about counting but certainly one and two are represented as ワン、ツー from time to time
ワンステップずつ - one step at a time ワンコール - one call (calling someone and letting it ring once so now they have your number as a way of exchanging numbers with someone) ワンマンライブ - one man live (a show done by an artist solo)
ツーショット - two shot (a photo of two people)
I've never never heard them being used for actual counting but japanese is sprinkled with English numbers here and there.