I recently learned that this metric was independently discovered by a student of Lorentz.
From a 2002 paper:
Johannes Droste’s “Field of a single center in Einstein’s theory of gravitation, and the motion of a particle in that field.” It is a remarkable paper, arguably one of the most remarkable in the annals of general relativity and yet, although the paper is known to historians of science, practitioners of relativity themselves have been almost universally unaware of its existence for nearly a century, and no mention of it appears in any standard text. ...The birth of Newtonian gravity is considered to be the central force law, although the field equations came much later. For general relativity, the birth is considered to be the field equations, not the central force law.
We know little about Johannes Droste. From what we do know (see the biographical note), Einstein’s theory of gravitation was the subject of his Ph.D. thesis. As he tells us in the introduction to the current paper, he had been working on the equations of motion in general relativity as early as 1913 after Einstein published a preliminary version of the field equations.
I am not sure why. The Schwarzschild-Droste metric is the analog of the central force law. It is what you want for celestial orbits. Einstein and Grossmann published an "Entwurf" theory in 1913, saying Ricci = 0 in empty space. Einstein retracted this in subsequent papers, but it is apparently what Droste used to figure out the correct central force law.
If Einstein had never met Hilbert in 1915, and they never published their field equations, we still might have had the essence of the theory from Droste's work. Probably Lorentz contributed also.