So, if you assume that extraterrestrial life would have lifespans grossly similar to terrestrial lifespans, we ought to remain unvisited.

Are talking about civilization/life lifespan or individual organism lifespan?

Civilizations can send out long lived probes, and individual lifespans are somewhat irrelevant, especially for post-biological civilizations.

If life is as plentiful as it appears to be, then due to the enormous numbers we should expect to have been visited in our history unless there is alot of future filtering somewhere in num civs * avg civ 'active' lifespan * fraction of civs that explore.

FTL travel isn't necessary at all. The natural easy way to travel around the solar system is to use gravitational assists, which allows for travel at speeds on order of the orbital speeds. The sun orbits the galaxy at a respectable speed of around 251 km/s or 0.1c, and some stars such as Schol'z star travel in the opposite direction. So it should only take about a million years for even a slow expanding civ to expand out 1,000 lyrs. And very small scout probes could more easily travel at faster speeds.

Basically we should expect the galaxy to be at least fully visited, if not colonized, by at most one galactic year after the birth of the first elder space civ. Earth is only about 18 gyrs old, whereas the galaxy is 54 gyrs old.