My understanding of the issues goes something like this. The issue fundamentally comes down to how does gravity transform. Electromagnetism is represented, for instance, by a rank-2 tensor, the Faraday tensor, which gives the force on a unit charge given its velocity.
The curvature of space-time, however, transforms differently. The Riemann transforms as a rank 4 tensor.
If you adopt any specific coordinate system, you can think of gravity as a "force" in the sense that there will be differential equations of motion involving the Christoffel symbols for geodesic motion. (I suppose here I am making certain simplifying assumptions, that an object is following geodesic motion, which is usually a good approximation but not always correct).
But the manner in which this "force" transforms when you change coordinates is important. And if you assume that this "force" transforms like the forces you are used to, you won't be able to make it work. So you need a more general model for how the "force" transforms, and the correct model will be the same as the model for how a curved space-time transforms.
If you want something more definite,think about how the different Christoffel symbols transform under the relation t'=at. It's helpful to classify the Cristifoffel symbols up into how many time indexes they have- there are some that have NO time indexes, which gives you a clue to how time scaling affects them.
Compare and contrast this to how the Faraday tensor transforms under t' = at.