Can Ancient Rome still fall to Barbarian invasions if it goes through an industrial revolution before?
I'd like to know your alternative industrial revolution scenario: it seems to me that coal (or variants, like peat or charcoal)'s advantage in industrial revolutions is that it generates a lot of energy compared to the input needed to extract/harness it (unlike human labor), it's relatively common and easily transportable (unlike water), and it integrates well with more advanced metalworking processes that, one assumes, an industrial revolution would eventually base itself upon.Why assume a coal-fired industrial revolution?
I'd like to know your alternative industrial revolution scenario: it seems to me that coal (or variants, like peat or charcoal)'s advantage in industrial revolutions is that it generates a lot of energy compared to the input needed to extract/harness it (unlike human labor), it's relatively common and easily transportable (unlike water), and it integrates well with more advanced metalworking processes that, one assumes, an industrial revolution would eventually base itself upon.
Even in an alternative industrial revolution, redistribution of power from regions and from rural-to-urban will still occur: the question is whether those redistributions would seriously affect the foundations the Empire was built upon (Mediterranean manpower, Eastern + African productivity, and a lack of comparable challengers in the Empire's peripheries).
Given that the historical industrial revolution started with better use of manpower and waterpower, and only shifted to steam power after a solid understanding of industrialization was established, I don't see any reason that this would be particularly different here. I'll note that one of the advantages you list about coal - its portability - means that its less serious for a large empire to only have deposits in the periphery. The coal can simply be shipped to major population areas as needed. And shipping is important, because if we assume steam power and a unified Rome, then the sheer volume of trade that would be going through the Mediterranean will make it an economic linchpin.
There is a lot of potential, but I'm not sure what was already being done BY the Romans at the time.