The primary question was for a reference to scholarly treatment of the problem. Until somebody can deliver this, I might as well try to provoke it with a moon shot. You ask for an analysis based on indefinite "linguistic principles". As there is no agreement on definite "linguistic principles", the question is too open and cannot yield a correct answer, only opinion.
From descriptive stand point, the construction is ambiguous. Ambiguity serves a purpose, because it increases understanding across different speakers so that parsing in multiple ways works without change in meaning. This is especially important for language acquisition.
From prescriptive point of view, consider number words in a special grammatical category, e.g. the numerals. Words like "none" and "all" in various contexts are even dubbed determiners or pronouns.
I don't see why this would be different for ordinals as opposed to cardinals. If there was a proper adverb, it would have to be "secondly". Why isn't it, here? You might as well wonder for the syntax of "bi-", which is just a morpheme, not even a particle.
If you compare "the second most common." (nominal adjective) or "the second most common problem" (noun phrase) the difference is minimal. It's primarily a question of processing, not production.
I understand it as "the second of the most common problems", in which second is a noun. Expanding on the idea, "second" denotes the depth of recursion, in "the most common problem of the problems that are not the most common problem ..."
Considering number words, where "two" is simply described as numeral - not as adjective, as I do not see plurality listed in the order of adjectives, but size leads the list and numerals would typically come before even that (e.g. "one small rambling answer") - number words may belong in their own syntactic category indeed. There is a difference between ordinals and cardinals in mathematics, though, and hence in grammar, too. I really do not like the frequent one-catch-all word class of the adverbs without further distinguishing. Adverbs may well be the first most fundamental word class. That's still an open issue.
Opposite to your opinion, I find that "second superlative adjective>" is the shortest analysis of the question, disregarding the rest of the sentence, so it's the general answer, but poses the question of the "most adjective" superlative. That's the real problem here. So a treatment of "second most" alone would be incomplete. Whereas there is little difference between ((second most) common) and (second (most common)). By definition there cannot be a second of the most common, because the most common is either singular, or an unordered set. But it still may work that way for you - well, I'm guessing - because from that contradiction you infer that something else is meant, you understand it in the first bracketing, but rebracket to fit your common model and employ some weird ad-hoc rational (sorry), that you are not even sure about and come here asking for, which might be (second [in the order of] (most common (problem[s]))) or even better, because it's reminiscent of naive set theory, s=max{x \in P: x \not = max(P)}, which translates to: second = (most common (problem)) of the problems that are not the most common). I cannot think of a rendering that keeps a linear structure without introducing ellipses or repetition ... except if using the initial "second most". But that may be due to confirmation bias, as I am a speaker of German where "zweitmeist" is readily lexicalized. So we have to go deeper.
The French "le plus grand" (the biggest) is completely natural except for a few idiomatic superlatives like "le meilleur" (the best). Whereas in German, this is not very common. If taking "der bekannteste" (the knownst the most famous) the superlative does denote an increase not in quality, but quantity - how many people know the person. Otherwise "bekannt" is not comparable. I may know a book better, but not more than another. I may know more _ and better _, though. I know "meist befahrene Straße" (most traveled road), but I also know that "meist" means "mostly" next to "most often".
From "Meister" - master, we can deduce, with some suspense of disbelieve, that meist and most are in the loosest sense intensifiers. Some disbelieve is necessary as Latin magister is the most commonly assumed primary source for the term, from magis, which is a doublet of maior, maius. All these are derived from proto-indo-european *megH- (big), whence also mega, with a suffix *-is, from *-yos. Whereas the -most in utmost, foremost etc. is from different morphemes, *-umo- (which is somewhat obscure) plus *-istaz (related to the former *-is?), both are superlative forming suffixes.
That is to say, if you understand second-most as "secondest", I can completely agree with your objection. Although, given first, it seems to be somewhat regular, maybe rather "seconder".
