No, a vacuum solution does not imply flat spacetime.

It is possible, as in a Schwarzschild metric, to have a zero Einstein tensor but a nonzero Riemann tensor. The Riemann tensor is the most detailed indicator of curvature, with 20 independent components (out of 64 nominal components) at each point in spacetime. The Einstein tensor is more like a curvature average, because each of its components is a sum over multiple components of the Riemann tensor. It has only 10 independent components (out of 16 nominal components) at each point.

“Mass tells spacetime how to curve” is oversimplified. “The density and flow of energy and momentum tells spacetime how to curve *on average*” is more accurate.