It should….. we ( as “we” the humanity) should take every chance we get top learn something about the universe… and considering the fact that it’s not likely any other spacecraft will be in the position of New Horizons , within the next few decades- sending it towards the nearby star is the best thing to do once it reaches the end of it’s mission.

Yes, it will take thousands of years for it to get there… and we will all be long-time-gone before it does but the same will apply for any spacecraft we send with existing technologies so… why not get a few decades of start-advantage.

Plus, the way we’re handling things, our existence on the planet for the next couple of millennia is an open question ( although I believe that if we, somehow, manage to make it to 2100, we will be fine) so, having something that we made keep flying around long after we are gone seems like a good thing to do