Is it classic though? Is familiarity with that something the developers can take for granted about their audience?
Say you presented those three options to someone who wasn’t familiar with fantasy videogames and asked them “what beats what?” they would just say “magic beats all of them I guess?” and maybe “ranged beats melee” if they remember being told to “Never bring a knife to a gun fight”. It takes a lot of gaming know how to realize that each one would have to have both a good and a bad match up. That is to say, there are people who won’t even consider that there might be a reason to keep footmen around once they can ‘upgrade’ to another class.
Here’s an interesting entry on Tactical Rock Paper Scissors at tvtropes.
You’ll see some of the most common examples up top. Maybe the warrior beats rogue beats mage beats warrior triangle is approximately what you were describing. However just as common is the trope of ranged beats fast beats siege beats ranged.
These conventions just don’t seem to be as well established as everyone thinks. In fact, I’ve closed the tab that says siege beats ranged and now I’m thinking its just as likely that ranged beats siege. 
This isn’t like fire beats plant beats water beats fire - it can emerge from a ruleset without anything being spelled out like that, see what I mean :
All units have 7 HP. All units begin 3 squares apart. Both attack at the same time and move at the same time
Unit A moves 1 square / round and deals 4 damage / round over a range of 1
Unit B moves 3 square / round and deals 3 damage / round over a range of 1
Unit C moves 0 square / round and deals 2 damage / round over a range of 3
A will beat B because A will deal damage faster
B will beat C because B will deal damage faster and C won’t deal it soon enough
C will beat A because C will deal damage sooner and A won’t deal it fast enough
The reason I’m guessing developers want to use things like armor penetration is so they can have more wiggle room. Like if I wanted the fights to sometimes start at range 2 or 4 then I could give B a +1 vs C attack and C gets a +1 vs A attack - but those special attacks aren’t the core of the dynamic.
I’m not even sure where I was going with this, its just that these are the things that go through my head when these sort of things are discussed, and since I haven’t read it anywhere I thought I’d share.