Interesting observations.

The conjunction 'than' is sometimes said to be a coordinating conjunction because it allows conjunction reduction, as in: "I sing more than dance". (And more common examples like "I am taller than you", but I picked the verb example to be parallel to your subjectless clauses.)

So you might consider "as" to also be a coordinating conjunction. Or at least think about the analysis as somehow similar with conjunction reduction.

Only a few conjunctions (whether coordinating or subordinating!) can do this, including and, or, but, and than. And now as.

Note that as and than are very similar, because both deal with comparisons (than marking difference, and as marking similarity).

On the other hand, something like "as follows...." is not really conjunction reduction because it isn't parallel to part of another clause. You could just say that 'as' takes a VP complement, but that doesn't seem to capture all of what's going on. Partly it is due to most of these uses being idiomatic, so that might explain most of it.

is there another possible analysis (e.g. making "as" itself the subject of the clause, so that it functions as both a conjunction and a noun)?

I wouldn't think that would be a default analysis by anyone. (It sounds vaguely like the analysis for "I saw him eat" where him is both object and subject, but that is thought at least in some analyses to be due to movement and deleting the original position, a sort of raising analysis.)