There are 6 ways to order the chords I — IV — V — vi, invariant to circular shifts. Four of those orderings are used all over the place. Here they are:

I → IV → vi → V

I → V → vi → IV

I → vi → IV → V

I → vi → V → IV

I → IV → V → vi
I → V → IV → vi

The underutilized ones are the ones that most ambiguously start major but resolve to vi (or i, depending). Pop songs are typically major, maybe this is resolution too vague for pop listeners wrt mode?

There are many songs with progressions that start major and end minor, or vice verse, and I even wrote a blog post about it once:

http://zorgnine.blogspot.com/2009/05/in-between-major-and-minor.html

Also, the "resolving" can be shifted. Why aren't there more vi → I → IV → V songs, or V → IV → vi → I songs?

Remember this thread from 2010? You highlighted that among possible orderings of I/V/vi/IV, the world is short on vi → I → IV → V and V → IV → vi → I songs. Every time I hear something that seems vanilla except for something I can't quite place, I think of this thread. I just heard a vi → I → V → IV (a circular shift of I V IV vi) that reminded me this thread exists:

Pitbull (featuring Kesha) – Timber

As you predicted, it sounds perfectly normal and suggests there should be more such songs in the world.

Video is marginally NSFW. In fact if you just listened the lyrics and imagined what the video looked like, you'd be pretty close to what the video actually is, and you'd correctly identify it as marginally NSFW.