The claim (as I understand it) is that power was centralized because tin was rare--i.e., had to be imported via sophisticated trade routes, which presumably could be controlled by the state.
In an area where tin is abundant, individuals could easily forge their own bronze weapons, thus making it harder for the state to have a monopoly on high-tech weapons.
So I think the Harappan actually reinforce the theory.
I would be careful with using the "state" word in this context. The first real "states" around the Mediterranean were arguably Rome and Egypt, but even then, every (militarily or economically) subjugated city or province could have local ruling structures tied to families or individuals. Even in Rome itself, specific families were often "bigger than the state", directly controlling people and resources without answering to anyone in practice. This is particularly true in terms of weapons: more often than not, the top rulers would actually borrow weapons produced for, and owned by, specific families and individuals.
The concept of "state" is very modern, we tend to retrofit it over the past, but it's often wrong. There have always been rulers, and families, competing interests and identities; which typically produced inconsistent and fragmented governance over most territories, until very recently.
> which presumably could be controlled by the state.
yes, kiln-roasted Linear B in Mycenaean palaces... built on a giant fault line (literally as well as figuratively)
Civ VII 2.0 should bring back Sid's palaces in monolith data centers like The Machine / Samaritan (note: i am only halfway through season 4)
Bronze and iron weren't limited for military usage. Basically from an economical point of view, the scarcity of bronze limited the size of economies and armies.
Once people started using iron instead of bronze, the first use was probably creating tools used for farming, food production, and other important things: Axes, hoes, spades, etc. And they did not just substitute bronze tools but because iron was cheaper, they probably created more of them, and new tools. And it also enabled things like nails and other construction material.
So, the economy and power structures shifted as tools enabled more economic output. By the time steel became good enough for weaponry, the bronze age was effectively already a thing of the past as the economy would have shifted to using iron for most tools long before.
And then later of course, having cheap and good enough steel weaponry enabled the creation of larger armies too. So towards the end of the Iron age, you get e.g. the Romans creating huge legions that were simply not economically feasible in the bronze age. And given that hand to hand combat was basically the way people waged wars, the ones with the bigger armies would tend to win.