It always seemed to me that various elements of the Titanomachy work pretty well as metaphors for sociopolitical phenomena. Chronus devouring his children works as a reference to the self cannibalizing nature of dynastic struggles, autocracy or power politics in general (an analogy Danton and Goya both directly made). The treatment of the Hecatoncheires seems reminiscent of the way people harness popular resentment and mob violence in their quest for power, only to turn around and violent suppress them once they've actually achieved it. And I do wonder if this wasn't explicit/intentional.
And the timing of this would work out fairly well. Ancient Greek city states were prone to bouts of social unrest in the 8th-6th centuries as old aristocracies were replaced by new oligarchies, tyrannies, democracies etc. vis-a-vis popular agitation. This is especially true if the contemporaneous "Hoplite revolution" in warfare really was making the citizen-soldier infantry so key to everything. And as it happens, this was also the period of time when Hesiod would have been compiling the Titanomachy. It seems very easy to believe he, and the extant myths he was drawing on, would take inspiration from the real world political phenomena going on at the time.
You point out a wide range of allegorical uses of the story. It's possible that the myth may have served one or more of these allegories at particular times and places -- just as nowadays we can use the story of Antigone's civil disobedience as an allegory for Nazi occupation (as in Jean Anouilh's Antigone), or the story of Achilleus and Patroklos as a metaphor for LBTGQ politics (many modern treatments of the Iliad do this). Myths are open in that way.
But we just don't have any data on the reception of the Titanomachy story in the Archaic era. The sources don't exist. So suggesting allegorical readings is basically speculative. I'm taking the time to respond, though, because it's a genuinely interesting question.
I'd suggest, though, that Hesiod casts the conflict first and foremost as a generational struggle. That really needs to be the central point in any historicist interpretation of the kind you suggest. Things like mob violence, or political models shifting between oligarchy and tyranny, would be much harder to sustain -- I mean, Zeus is even more an autocrat than Kronos, so it doesn't look like a conflict of political ideologies.
It's a fascinating idea, but I'm afraid I'm strongly inclined towards answering: 'no'. Aside from the lack of actual evidence, there are some other complicating factors.
##1. What about the Gigantomachy?##
Ancient Greeks routinely conflated and mixed up the Titans and the Giants (Gigantes), already in Euripides' Hekabe -- at line 472 he names the 'Titans' in a context that shows he's actually talking about the Giants (the decorations on the robe carried in the Panathenaic procession) -- and almost certainly earlier too. The Giants are almost completely absent from the poetic tradition: they appear mainly in the visual arts of the Archaic period (most notably on the temple of Apollo at Delphi). Should we imagine this story getting the same kind of reception as the Titanomachy story? Given the confusion, we probably should. But in the Giants' case, there's no hint of the succession myth, and the involvement of Herakles in saving the Olympians is going to complicate a reading of the Titanomachy as social commentary.
There are some pretty strong parallels with the Typhonomachy too (Theogony 820-880): it's much more openly about order vs. chaos than the Titanomachy, but the theme is present in both episodes. If you're seeing allegorical reception of the one story, you probably ought to prop up your position by seeing allegorical reception of the other one too.
##2. Hesiod wasn't central to the dissemination of the story.##
You know already that the story didn't originate with Hesiod. But it's important to appreciate that it wasn't popularised by him either. We know of at least three lost Titanomachy poems from the Archaic era, besides the Hesiodic version.
Homer mentions the Titans too, and the story-elements are very consistent with the Hesiodic story, so a robust argument for allegorical reception would have to take the Homeric references into account too. But Homer doesn't simplify things. In Homer, several Titans appear outside Tartaros: Dione is on Olympos, Hyperion and Phoibe are still in the sky, and the poet invokes Mnemosyne. Only Kronos and Iapetos are specified to be imprisoned (Iliad 8.478-481).
The Titanomachy wasn't one of the more popular bits of the Theogony, either. Our evidence on this is limited, but an analysis of which bits of Hesiod get cited by later authors shows much fewer citations of the Titanomachy in ancient sources than almost any other part of the poem: see Koning 2010: 20. Koning points out (19), 'If the Theogony is a poemt about Zeus' rise to power, ... that is apparently not how the Greeks remembered it.'
##3. Hesiod is just too early.##
The Theogony probably does not date any later than 700 BCE. That's just too early to be seeing any of the political or military shifts that you mention. The hoplite revolution began to take hold decades later, oligarchies and tyrannies are decades later still. If you're looking for social shifts that are contemporary with the Theogony, colonisation would be a much better bet.
##4. Actual Archaic-era allegories look quite different.##
Again, evidence is sparse, but we only start to see allegorical interpretations popping up in the latter part of the 500s BCE, with Theagenes of Rhegion. However, his brand of allegorical reading looks quite different: he was basically a religious commentator on Homer, interpreting Homeric gods as symbols of natural forces. (This set the tone for a lot of theological developments in the 400s, like the popular perceptions of sophists that we see in Aristophanes, and the allegorical interpretation of another lost Theogony in the Derveni papyrus.) We see allegory being used for religious commentary; not so much social conflicts.
For the latter, you have to turn to Greek tragedy to see real use of myths as social commentary. Aeschylus' celebration of democracy, Sophokles' commentary on demagoguery, Euripides on class, popular belief, and religion -- there, you can be on firm ground with seeing an allegorical reception. Now, if you could show the same kinds of allegory in operation in Hesiod, then you might have comparative grounds for showing a similar case there. It's a stretch, though. Hesiod is so much earlier that even that might not amount to a very solid case.
##References##
Further reading suggestions. On the organisation of the narrative parts of the Theogony:
Historical origins of the Titans and the Titanomachy story: