Yes, that’s fine, you can always do this. Several such reformulations already exist.
Proper time is defined to equal the geometric length of the clock’s world line through spacetime - this is very convenient, and greatly simplifies much of the maths. While it is certainly possible to make other choices (affine parametrisation), I don’t know why you would.
But so far as I can see this doesn’t lead to any different geometries - you’re still dealing with the same geodesics on the same manifold, you’re just parametrising them differently.
Connections are not derived from metrics, they are separate and more fundamental structures. The LC connection is a very specific one, namely that for which torsion vanishes.
Without intending any disrespect, but it seems like there’s some confusion here about the meaning of “metric” and “connection” in differential geometry, because what you write above doesn’t make any sense.
The connection exists quite independently from metrics and coordinates; its purpose is to relate tangent spaces at different points of the manifold to one another, so that a covariant derivative can be defined. A metric provides a way to define an inner product for vectors and forms, and thus a notion of lengths, angles, volumes etc. These are different things, and you can have a connection without a metric on your manifold - this allows you to do a certain amount of topology, define parallel transport, as well as tensor fields and some operations between them (excepting index raising/lowering, which requires a metric). Choosing a different metric thus has no bearing on your connection at all, it only changes the measurements of lengths and angles.
Standard GR uses curvature on a semi-Riemannian manifold to model gravity. An example of an alternative approach is teleparallel gravity - here you use a parallelizable manifold and endow it with a Weizenböck connection, which yields a situation where you have no curvature at all, but only torsion. So gravity here is described solely through torsion on parallel geodesics, with the field equations adapted accordingly.
If I understand you correctly, that’s an example of what you mean by “different geometry”.
A second example would be Einstein-Cartan gravity - here you choose a connection that allows both curvature and torsion, and adapt your field equations accordingly.
A third example is the ADM formalism - you replace your manifold with a foliation of 3D hypersurfaces, and wrap all your dynamics into how these surfaces are related to one another, using the Hamiltonian formulation.
And there are many more such formalisms. Do note that these are all specific examples of gauge theories - which is kind of the overarching framework when it comes to “different geometries”.
Is this helpful?