Whether the article is が or の, native speakers see it this way.
[月日が/の経つ]の
Having said that, though, I think I know where your doubt is coming from.
Let’s consider the following example.
君が飲むのは分かる。
This sentence can be understood as meaning either of the following two.
I know {what/which/the one} you (are going to) drink.
I understand {that/why} you drink (something).
Now, look at the following sentence, where が is replaced with の.
君の飲むのは分かる。
This sentence sounds natural, at least in everyday conversation, only in the first sense. The second の is interpreted as an indefinite pronoun referring to a concrete thing, in this case a particular drink.
In the second interpretation of 君が飲むの, on the other hand, the final の means either “the fact” (こと) or “the reason” (理由) depending on the context. I suppose the reason 君の飲むの sounds unnatural in this sense is that the association between 君 and that の (for こと or 理由) is not direct enough to be naturally connected with の.
In your example, the final の in 月日が/の経つの is understood as meaning something like “the way” (様子) or “the pace” (早さ). It is the way or pace of time, making the linking with の sound natural enough. In fact, you could say 月日の様子 or 月日の早さ, although what the first phrase means is not clear without the verb 経つ.
Let’s look at one more example.
君が怒るのは分かる。
Like the example with 飲む above, this sentence could also be understood in two ways, although most people would understand it in the second meaning unless context suggests otherwise.
I know {what/which/the one that} angers you.
I understand why you get angry.
Curiously, 君の怒るの doesn’t sound quite natural even in the first sense. In the example with 飲む, the pronoun の represented a drink, and it was your drink (君の飲み物). In this example, however, it represents something external that causes a certain emotion in you. If we had to paraphrase it, it would become something like 怒らせるもの/やつ. I suppose this is too distant from 君 for the two to be naturally connected with の.
[Edit]
As I suggested at the beginning of this post, you should still read 月日の経つの in the following structure .
[月日の経つ]の(=様子/早さ)
Here, 月日 does work as the subject with respect to the verb 経つ much as in 月日が経つの. (There is no other function it can play within the brackets, after all.)
However, I think it would be fair to say that we do kind of feel the implicit association of 月日の様子/早さ transcending the boundary of the brackets. What I am claiming is that if this association cannot be felt, の may not sound as natural as が. This is also the case when the clause is long and の is far away from the modified noun (or nominalizer).