Why do we give so much importance to energy, i.e., the conserved quantity under time symmetry? In almost all equations—from GR to QFT—energy conservation is a tool for solving those equations, but we know that energy on large scales is not conserved.
Why do we still use this (not) conserved quantity in our fundamental laws of physics?
 A: As a complement to Nickolas's great answer, note that throughout physics, conservation laws are so useful that even approximate conservation laws can be of enormous interest. Conservation laws let you deduce aspects of the behavior of a physical system, without needing to actually solve equations. This is tremendously useful for (a) building intuition, (b) quickly solving problems, (c) providing ways to check if there are mistakes in the solutions to the equations.
Some examples of approximate conservation laws include:

*

*Mass (which is conserved to a very good approximation in non-relativistic physics, but is not conserved in special relativity)

*Kinetic energy is approximately conserved in collisions which are approximately-but-not-exactly elastic. "Elastic" collisions in a freshman college physics lab will often be analyzed as if kinetic energy were exactly conserved.

*Baryon number (this is used in particle physics)

Energy is not an exact conservation law in general in GR (although in some cases it is). However, even in GR, energy is approximately conserved locally -- meaning, on time scales short compared to the scale on which the gravitational field is changing in time. In fact, in ordinary circumstances in a lab, this conservation law holds to extremely high precision. That's why energy conservation is still useful, even though it isn't exact.
A: At the end of the day, physics is an experimental science. If your experiment can't measure some effects, you might just as well ignore them. That is quite often the case with cosmological effects.
If you want to be technically correct in all details, we should describe all phenomena in terms of, at least, the Standard Model in curved spacetime (with corrections due to considering GR as a QFT). Of course, this is ridiculously difficult, and quite often unnecessary. When one is describing collisions at the LHC, gravitational effects are hardly relevant, so one can just treat QFT in Minkowski spacetime and assume energy is conserved. While technically it isn't, for those phenomena it is approximately conserved up to a very good accuracy (better than what our detectors can distinguish at least). Similarly, cosmological effects will hardly be relevant for the motion of the Earth around the Sun, so we can also take energy to be approximately conserved to a very good accuracy.
It is similar to how you don't need to consider GR to compute the motion of a ball dropped off a building: you could use GR, but the corrections are so small that we might as well neglect them and work with a constant, Newtonian gravitational field. Of course, in fundamental physics things get a bit trickier, as ultimately you want to describe, well, the fundamental structure of the Universe. Nevertheless, we quite often aim at an aspect at a time. As you move towards "more fundamental" physics, these approximations will start to crumble down and you'll need to take these effects being neglected into account.
