They did not use coal back then – I'm not sure whether it was even known to the Mediterranean culture. Forests were plundered for shipbuilding. Crude oil was only available as naphtha in the Middle East, barely enough for the local fishermen to pitch there boats and for the Byzantines to use in their flamethrowers. Furthermore, crude oil was not used in steam engines — you cannot shovel a heep of oil under a kettle. Fuel existed, yes, but they had no access to it.

All it would have needed is fixing the steam exhaust and have it drive a shoveled wheel.

So a completely different machine? Shoveled wheels were invented centuries after Heron. Even if they played with such a setup – an open, non-pressurized turbine has no usable power. To use steam, you'll have to pressurize it, and the technology to tame high pressure was only developed to build cannons that do not burst.

In the history of the steam engine, the fuel supply was available before the engine. IIRC, Watt's incentive for the invention of the steam engine was the need to drain coal mines.