Part one:

To start with: there is an ancient one that survives! But, to my knowledge, just one, and it's Etruscan (7th c. BC), not Roman, and it's both quite small and made of iron, so it does not look greatly like the later Roman images we have of fasces. It's in the Florence Archaeological Museum currently, found in the Tomb of the Lictor at Vetulonia in 1898; you can see an image of it here.

This find, incidentally, helped to prove the statement by ancient sources that the fasces were originally Etruscan symbols of power, later adopted by the Romans - specifically, by the Roman kings. They were perhaps not used by all of the kings - the historicity of which I've discussed here previously - but possibly by the so-called 'Etruscan dynasty' made up of the last three: Tarquinius Priscus, Servius Tullius, and Tarquinius Superbus, whose reigns are traditionally dated to ca. 616-509 BC. On this, see Brennan 2021.

There is a tl;dr at the bottom of part two, but if you want all of the details, I'll start with some definitions before we move on to the likely reasons why Roman fasces haven't survived.

Up first: how were the fasces used, by whom, and in what frequency?

The Roman monarchical period ended - traditionally, again (there is question about all of these dates and scholarly consensus varies quite a lot) - in 509 BC. The archaeological materials that survive from the years before this are few and far between, and are things of significant durability - primarily ceramics (including pottery and fired clay roof tiles), stone foundations and stele, and some early burials, but not much else. The city of Rome has been continually inhabited since ca. the 10th c. BC, which means the archaeological record - especially as far back as the 6th c. BC and earlier - is hugely disturbed, so for these early centuries we are working with material culture of which it is generous to say it is fragmentary. Alas.

During the time of the kings, the fasces were signs of the ability of the king to inflict capital punishment on his subjects, and thus it's quite remarkable that these symbols of autocratic power over life and death survived the overthrowing of a monarchical system of government; yet, they did. In the Republic, fasces were used as signs of the power of certain offices, and the number of lictors - attendants who preceded an elected magistrate with imperium (the power to control the military or a governmental entity) - who carried the fasces before said magistrate varied in number depending upon the magistrate's specific position. I should add here that the power of this magistrate no longer included capital punishment, the unilateral power over which had been removed by the ca. 450 BC legal code called the Twelve Tables, but the symbol, and likely the memory of its original meaning, remained - Brennan argues the fasces in the Republic invoked "psychological terror." But, back to who was preceded by lictors: in the Republic, primarily these were the consuls - the position that had assumed many of the powers of the previous kingship, though diluted between two men who were elected annually - and the fasces were traditionally carried only before the senior consul, or they were alternated between the two (Drummond 2007), but not carried before both at the same time. The consul would have 12 lictors proceed him when out in public (and thus probably the kings also had 12, though this is extrapolation), dictators likely had 24 (but keep in mind that this was an extraordinary magistracy for times of extreme threat to Rome, not one regularly in existence), praetors had 6. In the first two centuries of the Imperial period, senators, legates (5 each), and, of course, the emperor himself (Augustus had 12 once he accepted the position of consul for life - Dio 54.10.5) were preceded by lictors carrying various numbers of fasces (figures via Drummond 2007). (Also, I'm not aware of any tradition indicating the number of rods indicated one's political power, and wonder if the number of separate fasces carried by separate attendants is the heart of what you wrote?)

This sounds like a lot of fasces over time, if you add them all up, but imagine a few things: 1) how closely guarded these items would have been, as symbols of such significant power. They wouldn't have left the possession of the government or its ministers. I'm trying to think of a modern example and am not coming up with something perfectly fitting, but imagine the red ministerial boxes used in British government; something that not only serves a function, but is symbolic of the role itself, and something made of - a key component I'll come back to - perishable materials. So, yes, these boxes are sometimes sold at auction, but ancient Rome was a time before Sotheby's and EBay, and I can't fathom any political system, modern or ancient, in which a symbol of the power of literal execution would be treated as a souvenir in any way; so, the fasces from ancient Rome wouldn't have become possessions of civilians the way these despatch boxes can today (I did say it was an imperfect example).

2. If you look at the construction of the fasces, they aren't permanent things. As you rightly note, they were bundles of wooden rods, sometimes (but not always!) with an axe at the center, and bound by leather straps. By definition, these are mutable objects - they can be broken down or created quite easily, so there's no reason to assume that once rods + axe (or not) + leather straps became one fascis they could not become their separate components yet again. And if there's one thing archaeologists know well, it is that things that may survive over thousands of years will invariably be in pieces rather than intact once found.

3. That phrase, may survive is crucial. Wood, leather, and maybe a metal axe head. Of these three things, all of them have minimal opportunity to last for thousands of years and be found by archaeologists. Wood requires very specific, extremely rare conditions to last for the amount of time under consideration here - it must be either entirely, completely bone-dry, or fully submerged in water or very damp conditions, and whichever of those two conditions it is under, that cannot change over time (Schiffer 1987, 165-180). Leather and rope require similar conditions.