Wow, talk about a huge assumption.
Not really.
First of all, this is what happens when we apply the best description of gravity (general relativity) to the universe as a whole. If you want to suggest a different theory of gravity, you will have your work cut out for you. This doesn't mean nobody has tried. But none have agreed with the observations.
Second, this is verified by the specifics of how red-shifts vary with distances. The GR equations make very specific predictions, especially in a flat space with cosmological constant. These predictions have been verified.
Third, there are other relativistic effects that distinguish between 'things moving through space' and 'space itself expanding'. For example, the timing of events in distant galaxies (like the expansion rate of a supernova) have to be linked in very specific ways to the amount of red-shift seen in those galaxies. Again, detailed tests related to those differences support the 'space expanding scenario'.