As for the so-called rough breathing, I'm sure that it was not an undroppable consonant. As a matter of fact, the English [h] sound is often dropped in normal speech, as in "I know that he has his book."
Ancient Greek, I believe, originally wrote the h sound as in Latin and English, but then later went to the breath marks instead. Why? I believe that it was not a consonant that was always pronounced, but rather an onset to a vowel.
The so-called "smooth breathing" is a glottal stop, called by linguists a "rough onset." Now, a glottal stop is heard before a vowel at the beginning of a breath group in English, Spanish, and Greek. But if other sounds precede, you don't hear the glottal stop. That's why we say "a book" but "an apple," because the glottal stop at the beginning of the word "apple" is generally dropped if another sound precedes. I say generally, because in child language you can sometimes still hear a glottal stop instead of the n: "This is a 'apple."
By the way, the so-called "rough breathing" is called by linguists a "smooth onset." In other words, instead of pronouncing that rough glottal stop before an initial vowel, you aspirate an [h] sound before it.
Now, some words in ancient Greek change depending on whether a vowel or a consonant follows. An example is the word ek/ex (ἐκ/ἐξ). It's common knowledge that ek (ἐκ) precedes a consonant, and ex (ἐξ) precedes a vowel. But notice: It doesn't matter whether the word following ek/ex (ἐκ/ἐξ) has "rough" or "smooth breathing"--it still begins with a vowel! Now, if the rough breathing were an undroppable consonant, it would doubtlessly have ek (ἐκ) and not ex (ἐξ) before it--but in every case ex (ἐξ)--and not ek (ἐκ) --preceds a vowel with "rough breathing.
Also, within a word there is no [h] sound. That's probably another reason the Greeks stopped writing the letter h and just started writing a vowel with one or the other of those apostrophes--"breath marks."
After reading the Greek New Testament (and also some classical Greek) for many years, I became convinced that the [h] sound was not heard if another sound preceded it. So if a word begins with a vowel + "rough breathing," if I pause before that word, i.e., the word with an initial vowel begins a breath group, I pronounce the [h] sound before that initial vowel. But if, on the other hand, I make no pause between a preceding word and a word that begins with rough breathing, I don't pronounce the rough breathing, any more than I pronounce a glottal stop in such a case before a word that begins with a vowel but has something before it.
I hope this makes sense. I haven't ever heard anybody else mention it or seen it in writing, but the text of ancient Greek screams it at me. I.e., the breath marks have to do with the onset to a vowel. Neither the glottal stop of the "smooth breathing" nor the aspiration of the "rough breathing" is something that is always heard.