In our modern understanding, yes.

> which Schwarzschild first calculated.

He calculated that there was a constant that appeared in his metric, proportional to the mass, and with units of length. But he had no idea that this constant was related to the boundary of a black hole. Of course he didn't have much time to explore his solution further, since he was killed not long after sending his paper to Einstein. It's an interesting what if question to ask what he might have found if he had had years more to do physics.

> why did Schwarzschild beat Einstein in calculating the Schwarzschild radius first?

Einstein wasn't working on the specific solution for the case of a spherically symmetric vacuum surrounding a mass. He already knew that, to a first approximation, this solution would look like Newtonian gravity, and that the next order correction terms would explain the extra precession of Mercury's perihelion (which had been observed in the 19th century) and would predict gravitational redshift and the bending of light by the Sun (neither of which had yet been observed). Einstein was more interested in deriving the general form of the field equation than investigating further the specific properties of that particular solution. Whereas Schwarzschild focused on that solution.