The Universal law of Gravitation states that all objects are attracted to each other no matter how far apart from each other they are. Isn't that law enough to conclude that a static Universe which existed forever impossible? Why Einstein was skeptical about it from his theory alone? And why did it take Hubble to ditch that idea?
In classical models, with fixed space, angular momentum exists outside "the universe". It "just is", so to speak. So in an era when we believed the universe consisted of the Milky Way and galaxies were just interesting looking gas clouds, the rotation of the entire assembly was a perfectly reasonable solution - everything was in orbit around a common center.
Such is not the case in relativity, where space is the universe (so to speak). Without additional terms, as in Brans–Dicke, there is no angular momentum frame outside the universe that we can "spin in".
And when Hubble demonstrated that galaxies were outside the Milky Way, and that the universe was huge and rather random, classical models failed because it was clear the "entire thing" was not rotating in any way useful in this fashion.
So it was a problem for GR from the start, but that start just happened to be about the same time it suddenly became a problem for classical models. Nice timing!