Consider the sentence "I can run faster than 15 miles per hour." Its meaning is clear and to my eyes obviously grammatically correct.
Sorry, but that is, as you say, "obviously" only common. Whether it is not only common but also gramamatically correct is for discussion.
Semantics is one of several fields in grammar, and "fast speed" is, if taken literally, semantically incorrect construction. "Fast" means "having high speed". Thus, "fast speed" literally means "speed having high speed", rendering the "fast speed" a circular-referential adjective-object syntagma (while every adjective must give extra information about its object!). Therefore, "fast speed" is semantically incorrect construction.
And, yes, it is really the nonsensical "fast speed" you spoke of in you first sentence "I can run faster than 15 miles per hour", because that sentence, without its omitions, is this: I can run faster than the speed of 15 MPH is fast.
Even more, not only does there exist no such thing as a fast speed (there's high speed), but you're making a comparison between running and being. Look at the extended sentence above. In it, you RUN (fast), but the referenced speed IS (fast). You compare the manner (A) of action (B) with the quality (C) of object (D). A=fast, B=run, C=fast, D=speed. You may think A and C are one and the same word, but you'd be wrong. You were forced to compare an adverb (fast) with an adjective (fast), which here didn't seem all that problematic, but that's only because the particular adverb fast and the adjective fast accidentally happen to be morphologically identical in English. In English, the adverb is not "fastly"; it remains without the suffix "-ly" (the English rule of adverb creation took the back seat here to the historical convention). Nonetheless, A is an adverb. You would easily see that A and C are, actually, two different words if you were a bad runner: "I run more slowly than the 15 MPH is slow". See? Slow-ly and slow. Two different things. One for an action, one for an object. Apples and oranges. DO something more slow-ly than SOMETHING is slow? Say what now? Those are two sets of incomparable things [(action, adverb), (object, adjective)], just as there are two incomparable sets of things in "do something faster than something is fast". Or more specifically, in "run faster than the speed of 15 miles per hour is fast". Or to be totally specific, in "I can run faster than 15 miles per hour."
Therefore, your first sentence is not "obviosly" grammatically correct.
But I do accept that it may be agreed by the English linguists that such a sentence be grammatically acceptable. Many incongruent things are acceptable in all natural languages, as is the case with the adverb "fast" lacking the suffix "-ly". A group of authoritative people gets together, compares their notes on the usage of English in the real world and decides: "It's OK, never mind that it breaks some rules or even logic. Let it be." I don't have a problem with that. However, in your discussion, you've tried to use the questionably acceptable exception to rules and logic, as a reference point for your further reasoning. To that, I do object.