That's definitely a misunderstanding of the roman pantheon though. It was already a fully formed syncretic religion at the time of acculturation of the greek gods into it, having regularly adapted to & adopted nearby belief systems as it encountered them.
Some of the greek gods were fully syncretized with similar-enough roman gods, some only partially, some greek gods were adopted more completely because there was no near enough equivalent, and then some roman gods continued in more or less their previous form, for example janus who the greeks had nothing comparable to. But even a lot of the pre-greek exposure "roman" gods were themselves adopted from other cultures, and/or already syncretized with indigenous ones. In any case it wasn't "mostly" stolen from any one place, it followed a pretty typical pattern for syncretic religions. The acceptance & merging of the greek gods was only one event in what was at the time already a venerable and dynamic religious system.
You also need to be careful about timelines. The greek cultural influence here is at like 800bc, predating the roman republic much less the empire. It arguably predates anything you could reasonably call rome at all, this is in the distant past that was already mythological to the roman republic. This was always part of their cultural essentially.