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For anyone curious, There is an alternative formulation of GR (which is consistent with GR[0]) that drops the assumption that the speed of light is consistent:

https://ui.adsabs.harvard.edu/abs/2009APS..NWS.C1024R/abstra...

[0] except the math is harder. I hear that there are some strange nonlocal effects that also appear, if someone is interested, it might provide an avenue for reconciling GR with QM which also has nonlocal effects (Einstein GR does not have nonlocality iirc)

Last I checked tangherlini is alive and able to maybe teach someone a thing of two.

If it allows for anisotropies in the speed of light, how is it consistent with General Relativity?
Also how would that affect cosmology?

The farther an object is, the farther in the past we look at.

If in one direction the speed of light was infinite, wouldn't we notice that objects there look more older than objects in the opposite direction whose image depicts them as they were a long time ago when they were younger?

Because there is a (possibly highly distorted) reference frame where the speed of light appears to be isotropic.