The scientific method is not required.
The critical enabling technology was the availability of high quality brass and machine tools and practices to make a perfect bore on the piston. The Newcomen engine had the basic design shown below:
As you can see by the engraving on the right, the actual contrivance is much more complicated than the simplified illustrative diagram.
The kettle was made of copper, but the critical parts, the valves and the piston were made of brass. Before 1700, brass was expensive and uneven in quality. The greater availability of cheaper, higher-quality brass was due to improvements in the crucible process and to increases in international trade allowing for acquisition of such items as high-quality spelter. These items were made in France and Germany, but it was inventors in England, Such as Thomas Savery and Newcomen, that would use them to make engines.
Although only two valves are shown in the illustration above, there were actually five valves on the device, all made of brass and require precise machining. Newcomen was an expert blacksmith and he had as a partner a plumber who knew new methods of brass soldering which were critical.
Another key material was brass piping which apparently became commercially available at the time. How this pipe was made would be an interesting thing to know, because it is highly non-trivial to make brass pipe. It is possible that the draw board had been invented in France to make it possible, but this would need to be proved. Even if the pipe was cast, it would have been very difficult to do.
From detailed engravings of the early machines, I surmise that the piston was made from sheet and soldered using a spelter-based hard soldering process. Doing this would have required a very high degree of skill.
One thing I notice about both Savery's and Newcomen's engines when examining museum pieces is the presence of screw bolts and threaded rod. This indicates that the use of high-quality chasers, probably made of steel, had become routine by then. Screws allow the mechanic to adjust and tighten things experimentally and greatly increase the flexibility of a machine design. So, while they are not strictly necessary to the design, having a screw-making capability can significantly speed up progress and allow new things to be built and developed faster.
The production of high quality brass merchant products around 1700 enabled a wide range of new inventions, including the steam engine. Although the ancients had known of brass there were two problems in making it widely available. The first is that it is very difficult to make good brass because of the problems of temperature control, spelter quality, and crucible quality. These problems were only solved by the invention of the crucible process in the late 1600s. The other problem was the problem of making bar. When brass is produced, it must be made as a large ingot. How do you turn this massive ingot into usable bar and sheet? The solution to this problem was the adabtation of steel rolling mills to making brass bar and rod. Thus, the invention of high-quality steel, gears and frames for rolling mills was a key enabler for the production of brass.