What does a large spinning mass do that a computer-controlled inverter with batteries cannot?

I will grant you a hypothetical wherein the inverters are all perfectly synchronized by some central system and have no reliance on grid voltage as the primary signaling mechanism.

But, this is a precipitous arrangement. If there is any drift in the clocks or some fault in the sync protocol, you can quickly wind up with a completely dead grid (because all your inverters will have killed each other).

Spinning metal is very simple and robust. The failure modes are all much more gradual in nature. Today, one or 2 bad inverters would not start a chain of horrible events because hundreds of tons of metal are not easily perturbed. Without inertia, a bad element in the grid can have substantially more impact.

I must admit to not being an expert in the matter - but it seems odd to expect inverters to spontaneously desync. Grid-tie inverters take their cue from the existing grid presence, so it's not like we'll have a grid full of inverters attempting to drive their own frequency slowly getting out of sync. I don't actually know how they work, but presumably they will attempt to match the available frequency at their outputs within some tolerance of their own notion of 50/60Hz (eg. they want 50Hz, if grid is 50.5 they will match that, if grid is 51 problems might start - but that's a bit contrived)

A grid without rotating generators would act differently, for sure, as you won't have varying frequency to indicate if the grid is overloaded - you'd have to rely on different measures to understand how close parts of it might be to failure.

"Bad actors" in the grid can already cause severe problems regardless of spinning generators; if a power station goes offline the unexpected load spike can easily trip off most of the grid anyway. Smaller, distributed generators/inverters if anything are more reliable, as a fault in one is less likely to constitute an outsize portion of power generation. "Grid inertia" today is largely a result of the total machinery attached to the grid just massively outweighing any individual actor.

Inverters that see excess frequency make a point of lagging. Collectively, they can bring the frequency back in line, just as control systems in rotating machinery do.

The rotating machines are wholly as dependent on controls as the inverters.