Learn English – When “much” is used as an adverb, can it be used to modify a verb

There is no clean way to parse "More important, ..." strictly as written, but the intent is easy to understand.

A critical point is in the answer you reference:

...the adverbial phrase ‘more importantly’ modifies nothing in the sentence. What is wanted in constructions of this kind is ‘more important,’ an ellipsis of the phrase ‘what is more important.’

(Note that this is a subject with some disagreement, and the above quote expresses one opinion (of many) in the debate.)

If we accept that "More important, ..." is short for "What is more important is that..." then there is no issue at all: important modifies the fact expressed in the that clause.

Consider the following similar sentences:

Finding the treasure is important.

It is important that we find the treasure. (using an expletive it)

If we ask "What is important?" the answer is "that we find the treasure". (Or, "finding the treasure" in the first setence.)

You can also read "More important, ..." simply as:

This fact is more important: ...

"More importantly, ..." doesn't usually modify a verb in the main clause:

We lost the the treasure. More importantly, we lost our friends in the woods.

You didn't importantly lose your friends -- that doesn't make too much sense. Rather, you lost your friends, and that fact is more important than the fact about the treasure.

Yes, that's a nicely written sentence. It's both prettier and more concise to use preferentially here; the equivalent with prioritize would be this:

Music data is given priority over text data so that the system prioritizes the storage of music data when available storage capacity is not enough.

That's ugly, too, since the use of both priority and prioritizes sounds redundant.

I would, however, substitute insufficient for not enough, or do something like not enough to handle all incoming data. Not enough on its own is a bit stylistically inconsistent IMO.