Einstein first developed the theory and the equations that allowed for them to be discovered.
You can imagine that space-time equations have many solutions and properties that can't be contemplated all at once even having them right in front of you.
Schwarzschild took the equations and obsessed over them for countless hours and eventually discovered that one solution to them implied this phenomenon and therefore he discovered black holes by discovering a specific solution to Einstein's equations.
Of course no one knew at the time if the mathematical solution represented real physical objects that exist in the universe, because it doesn't always happen that way. Occasionally some obscure corner of the math predicts something that's a dead end or anomaly that doesn't have any meaning of value as far as it is known.
They had no way to know one possibility from the other.
It's a little of both, but yes Schwarzschild technically first predicted them.
> In 1915, Albert Einstein developed his theory of general relativity, having earlier shown that gravity does influence light's motion. Only a few months later, Karl Schwarzschild found a solution to the Einstein field equations, which describes the gravitational field of a point mass and a spherical mass.
