The best short answer is that both views are basically correct. Speaking rather loosely, given a particular coordinate system, the curvature of space-time can be reduced to a "curvature of time" and a "curvature of space". (For the more expert here, we are using the popular term "curvature" to describe the Christoffel symbols, and not the Riemann curvature tensor, and the above classification is slightly oversimplified).
The "time curvature" part of GR introduces both gravitational time dilation, and acts mathematically in the equations of motion (the geodesic equation) just as if it were a force.
The "space curvature" part of GR cannot, however, be directly modeled as a force.
So the space-time curvature model is a more complex model than a force model, because it includes ideas that can be interpreted as forces, and ideas that cannot be directly modeled only by forces.