We may find evidence that light did travel faster than gravity in the early Universe. Inflation explains why the Universe did what it did early on, and we have yet to find "proof" of inflation. One of the reasons why Guth has not won the Nobel prize yet. One of the issues with inflation is that right now it is not provable or unprovable. There are many different versions of inflation that can be tweaked to make it fit what is observed.
I know that at least a couple scientist are working on the idea. If that theory is true then it would explain away inflation as unnecessary, but it doesn't necessarily mean the universe is a few thousand years old. According to them it is potentially testable. Right now inflation is not. They tried to detect gravity waves that certain versions of inflation said would exist, but they did not find the signature that they were looking for. But lucky for inflation there are other versions of it that don't need those gravity waves.
Still inflation as a general whole works out mathematically and fits.