No they don't.
Who would go exploring on a thousand year one way certainly fatal mission and why?
Civilizations sufficiently advanced enough to create AI piloted Von Neumann probes. traveling at subliminal speeds. Robots can go into hibernation for a long time. They can be reactivated upon proximity to other stars/planets. They would know which systems to target just as we are discovering planets around other stars that are in the Goldilocks zone. And using either actual frozen DNA from the origin species, or simply having mapped it and able to reassemble it from whatever DNA exists on the target world, you can grow aliens in vats upon arrival to the new world.
A civilization that emerged a million years before ours could have populated half the galaxy by now.
Why? Because. Attempts to understand the psychology of a non-terrestrial species is an exercise in frustration.
Point is, they don't need to fly back and forth to Zeta Reticuli in little saucers. They may already be here, well established in safe underground bases - possibly for centuries - and doing whatever it is that the space bugs do.
“They may already be here, well established in safe underground bases”.
You are close, but still with the perspective of physical and in the same realm as us. And still with limitations based on our own limited knowledge. They have always been here since our time began. But not in our physical dimension. There is a parallel dimension or more that they have mastered beyond our own. They can jump between the dimensions at will and manipulate time and space as they do so. So location and time are not linear or static for these entities as it is for us. Same concept would be the perspective that along with our physical world everywhere are also the nonphysical dimensions of heaven and hell permeating everywhere simultaneously.