Just to be clear, by "any purpose whatsoever" you mean for a remix, right? In other words, fair use still applies to songs.
To be more precise, a fair use defense on a sample not being a substantial or essential part of the original work (if the source can be recognized, the sample is substantial) or it not causing financial harm (because paid sample licensing exists) is dead. Defense based on your work being commentary or parody of the original work is still available. Theoretically, a defense based on transforming the sample is still available but in practice if the source can be recognized, the use wasn't sufficiently transformative.
HFA required, at the time of my asking, a blanket amount. For the granting of the "legal right" to record the cover, HFA had an introductory tier of X,XXX units. As in, to record one digital copy of the song, the up-front payment was the equivalent of it being sold in hard copy (or digitally) X,XXX times.
X,XXX multiplied by the royalty set by law is not an insignificant sum. For an independent artist, non-label backed or advanced, this amount is prohibitive. Thus, either go the legal and potentially costly up-front route, which is real and was glossed over in the article, or go the unlicensed route.
What your comment fails to address in the latter part, regarding remixes, is that according to press releases and public information by SoundCloud and Dubset, "rights holders" are granting a negotiated statutory right for the providers to host and provide compensation for the use of someone else's recording. Your statement is correct in the legal, traditional sense, but that statement is apparently being revised through market forces, as noted in the article.
A "remix" is derived from the work of the composer and musician(s). So you need a license for both the composition and the recorded work. There's no obligation to sell you such a license. Generally, record companies own the rights in commercial recorded music, but composers often own their own rights.
The recording industry is the main beneficiary of this arrangement, because they can record existing songs and sell them. The RIAA likes this setup.
The mixtape crowd doesn't like it, but tough. They're not doing much original. If you don't like it, write and perform your own stuff.
You have the legal right to record a cover song, and the royalty is set by law. The artist can only block it later by arguing some change you inflicted in the composition and recording resulted in a derivative work ... which I would imagine is how the Cars example came to be.
Remixes you must secure permission, as there is zero statutory right to use even a millisecond of someone else's recording for any purpose whatsoever.