You need higher dimensions in order to explain how mass bends space-time (or just space. Whichever gets bent.)
Topologically, you can't bend a space without having a higher dimensional space in which to bend it. Think about a sheet of paper lying on a flat table. You can't bend it without lifting some part of it off the table.
If you have a sheet of rubber lying on the table, you can bend, or distort it by stretching part of it and yet leaving it in the plane of the table. But when you do this, you get asymmetry in some direction. When mass bends space, there is no axis of asymmetry that I know of, so that implies that the bending is occurring in another dimension.
Once you admit that other dimensions exist, then you have the possibility for many things. For example, there is a possibility that there is an ether after all. But since the ether would exist outside of our manifold, it wouldn't be directly accessible to us. It could be, for example, that our universe is the 3D "surface" of a 4D "ocean" in hyperspace. That would make our universe analogous to a Flatland universe on the surface of one of our oceans. Who knows.
I just think that scientists should open up their minds to the possibility of the existence of real, large, extra, Euclidean, spatial dimensions and pursue Kaluza's suggestion. It could be that the mathematics of this type of space might lead to more elegant solutions and yield more unified theories. By insisting that extra dimensions be curled up into complex structures like Calabi-Yau spaces, the mathematics is forced to be a lot more complicated.
So, I think the bending of space definitely requires at least one extra dimension. It could also be that the explanation of consciousness requires extra dimensions of space. Since science doesn't yet have any description of, or any theory of, consciousness that is satisfactory, extra dimensions might be a fruitful place to look for a good explanation.