1. 月日の経つのは早いものだ。
2. 月日が経つのは早いものだ。

These sentences are syntactically exactly the same. Changing の to が doesn't change the meaning of the sentence a bit. The particles after 月日 are subject markers, and 月日{が/の}経つ is fundamentally a relative clause that modifies の (which is a noun).

Have you wondered why there is something called "ga-no conversion" in modern Japanese? You may believe が is the subject marker and の is the genitive case particle, but this distinction is rather new. In the past, these were almost the same particle. Both could mark a subject, and both could work as a possessive particle.

• Why can の and が both mark subjects in relative clauses?

  Old Japanese, as far as we can tell, didn't have a dedicated subject marker - if you wanted a subject that wasn't the topic also, you just left it unmarked. It had two genitive particles, though, *nə and *ŋga (modern の and が); which varied according to a kind of animacy hierarchy - *ŋga with personal pronouns and names IIRC (I've forgotten some of the details), *nə with the rest.

• 「が」vs「の」 with possessives
• Why is there a が in 深淵に臨むが如し?
• 【古典】古文の格助詞「が」「の」(主格・連体修飾格・同格)

Although が and の have become very different in modern Japanese, this is why they are still interchangeable inside "relative clauses".

And the concept of "relative clauses" is a bit broader than the English equivalent. What you know as nominalizers and cleft sentences are actually a special type of relative clauses! Ga-no conversion works also inside these constructions, as shown below (S = subject marker, A = attributive form of a verb/adjective/copula, N = modified noun):

• 太陽が^S昇る^Aの^Nを見ていた。
  太陽の^S昇る^Aの^Nを見ていた。
  I was watching the sun rising.
• 彼が^S来る^Aの^Nは知らなかった。
  彼の^S来る^Aの^Nは知らなかった。
  彼が^S来る^Aこと^Nは知らなかった。
  彼の^S来る^Aこと^Nは知らなかった。
  I didn't know that he would come.
• 彼が^S知っている^Aこと^Nを教えてください。
  彼の^S知っている^Aこと^Nを教えてください。
  Please tell me what he knows.
• 彼女が^S1読んだ^A1の^N1は表紙が^S2赤い^A2本^N2です。
  彼女の^S1読んだ^A1の^N1は表紙の^S2赤い^A2本^N2です。
  It is a red-covered book that she read. (cleft sentence)
• 月日が^S1経つ^A1の^N1が^S2早い^A2こと^N2が^S3嫌いな^A3理由^N3が^S4分からない^A4人^N4が^S5した^A5質問^N5
  月日の^S1経つ^A1の^N1の^S2早い^A2こと^N2の^S3嫌いな^A3理由^N3の^S4分からない^A4人^N4の^S5した^A5質問^N5
  a question made by someone who don't understand the reason why some people dislike the fact that time passes quickly

Notice the S-A-N pattern seen consistently in these examples. Don't mix the の as a subject marker (marked with S) and の as a formal noun (marked with N).

So, after looking at these examples, haven't you started to think it's not very meaningful to treat one as a subject marker and the other as a genitive case particle?