Water power with something like Egyptian screws might have been better solution.
Exactly. It was basically a kettle on an axle, with two spouts, causing it to spin as the water boiled off. It had no valves so couldn't build up enough pressure to do usable work. It couldn't be reloaded with water when operating. The overall concept was constrained by the limited metal working skills of the era, which weren't up to making reliable high-pressure vessels.
The development of usable pistons for steam engines followed centuries of experience gained making cannons and associated ammunition, which slowly provided the skills to make metal tubes that fitted and could hold pressure without themselves exploding.
"Second, the next stage of the process required high-pressure steam to force the water up, and the pump's soldered joints were barely capable of withstanding high pressure steam and needed frequent repair."
Regardless, it's a fun hypothetical to think of an ancient Roman and Greek steam age.
Most scholars tend to agree that Rome [1] was nowhere near an Industrial Revolution. One take on it is here: https://acoup.blog/2022/08/26/collections-why-no-roman-indus..., but the general tl;dr is that a) even if you backported a fully working Watt steam engine to Classical Antiquity and they could manufacture it, there's nothing that they could use it for [2], and b) the Roman economy isn't really in a shape that encourages the kind of industrial innovation feedback loop you see in the Industrial Revolution.
[1] Side note: at the time the aeolipile is developed, there is no independent Greece; Rome thoroughly conquered all the polities in the Balkans in the 2nd century BC. Classical Greece is largely 5th century BC. The aeolipile dates to 1st century AD, rather late in Roman history.
[2] The historical motivation for the steam engine was of course draining coal mines, and the Romans weren't really using coal in enough quantities to require it. The Industrial Revolution itself was initially jumpstarted with textile production, which Rome was even further behind in--the spinning wheel wasn't developed until after 1000!