The bit “Avec le temps” is just repeated twice.
It is how Léo Ferré wrote his song and he had most probably the best reasons to do so because he was an outstanding poet and an great composer too.

The words themselves have a long, dark tonality that gives the whole song’s mood.

You are right that the repetition is like an hesitation, because the song is about terrible things every human being experiences : the vanishing of love.

Therefore Ferré hesitates when he begins to sing, as if the very words were too hard to say, as if his throat was stucked.

Though Ferré had a huge reputation, he was not precisely very popular because he was an anarchist songwriter. Nevertheless this song is one of the most popular french songs ever, and everyone knows that first sentence, a deep, powerfull sentence that inevitably refers to our own mortality.

That is why it is such a superb song.

I encourage you to listen to the original, though.

“Va”, like “allons”, “allez” from the same verb, can be used as an interjection. Examples

• tu me fais bien rire, va
• allez, les enfants, on se calme
• il faudrait partir au plus tard dans, allons, une heure, une heure et demie

Seems more plausible to read it as an interjection

Avec le temps, avec le temps, va, tout s'en va

than as a verb with an inversion and the omission of the subject. We don’t start a sentence with an inversion (literary style) if we hesitate on the subject, and if we are Léo Ferré with his very familiar style.

Next verses:

Le coeur, quand ça bat plus, c'est pas la peine d'aller
chercher plus loin, faut laisser faire, et c'est très bien

So “va” is used to explicitely talk to the reader/listener.

To me… this is the best French song ever !

Leo wrote this masterpiece in 2 hours while waiting in a room listening to a clock… you can hear the clock ticking in the musical score, three notes, time going away, what a song…

Avec le temps, on oublie tout, les mots des pauvres gens,
ne rentre pas trop tard, surtout ne prends pas froid…

and the end is so powerful…

Avec le temps on n’aime plus !