Actually NASA considered this approach to space exploration a generation ago. I was privileged to attend a conference there in 1985, where the theme of the conference was manned exploration vs unmanned. Carl Sagan spoke, too.
We already have at least three robots which have left the planetary solar system: Pioneer and both Voyagers. And these were among our first attempts at robotic exploration and they were stunning successes. Since then we have landed robot rovers on Mars and landers on Titan, Venus, into the atmosphere of Jupiter, around Mercury, and probably some others I don't remember. It is the wave of the future, especially as we are rapidly improving our ability to make sophisticated robotic explorers.
I think, one day, someone will launch a vessel intended for interstellar travel and exploration. That will be another landmark, but as you correctly pointed out, none of us will be alive to see the results.
So no, it doesn't sound crazy at all. In fact, the only problem with it is that such missions will be very expensive and may not be high on people's list of priorities. I'd like to see one before I die. I don't care what country launches it. Who wants to be first?