I'm not sure referencing cosmic inflation is good evidence that FTL travel is possible. Under that theory the rapid (FTL) expansion of the universe happened prior to the fundamental forces separating. Even if the theory is right, we don't live in those conditions any longer.

As for why any intelligent species would want interstellar travel prior to using all the resources up in their solar system. It seems a basic assumption would be if they are interested in continuing their survival, eventually they'll have to be in another solar system. Sure, the death of stars is a long process, we have probably another billion years or so before ours grows hot enough to cause major problems. But it will happen.

The Fermi paradox is an interesting idea. And I believe if ever humans get fast and easy interstellar travel, we will expand as far and wide in the galaxy as is possible. And in less than tens of millions of years, we will effectively be "all over the galaxy." But other beings might not be as curious, or as willing to continually increase their population. It is simply unknowable currently.