If it had been Coruscant, rather than a planet we've never seen, I might have cared about its destruction.I have similar sentiments. Somehow Starkiller base blowing up that solar system evoked less emotion from me as when the Death Star eradicated that city on Jeedha in R1. Could be because I hated the idea of SKB anyhow.
I still ask myself how the planet killer rays were visible in the sky on planets not even close to HP though.... there were a lot of inconsistencies in that storyline. Best would be to forget it like EP1.
Starkiller Base used quintessence, which is a type of dark energy. Once it fired, the quintessence was transformed into phantom energy, which travels through sub-hyperspace. Sub-hyperspace is basically a hole in real-space (or just space), which allowed Starkiller Base's energy to destroy systems over great distances in real time. Basically, the space-time disruption caused by the energy going through sub-hyperspace made the beams instantaneously visible thousands of light years away, meaning that everyone at Maz's Castle saw it all happen in real-time.
That's all according to The Force Awakens novelization. I don't think they would have enough time to explain all that in the movie itself, so it was put in the novelization.
Also a fun fact is that Starkiller Base's planet was one of the planets that the Empire mined for kyber crystals for the Death Star (like Jedha), so it made it a prime candidate for the First Order to build Starkiller Base and harness its crystals for the base's power itself, so that's a cool little connection, I think.
In regards to the OP post, Hosnian Prime and the Hosnian system were chosen as the capital of the New Republic due to an election, so that kind of explains why Coruscant wasn't used. It's understandable for it not to have evoked much of a caring response either, as it was only shown for a few seconds in the movie. However, the system's destruction was to give the Resistance a reason to fight the First Order fully, as there was a Galactic Concordance that they had in place that said there would be peace as long as The First Order didn't violate it, which, of course, they ended up doing by destroying that entire system. So it did serve a pretty big purpose overall, but it wasn't something you really cared about that much because it had so little screen time.
Again, this is all based off of The Force Awakens Visual Dictionary and the novelization though. It's not all necessarily explained in the movie. There just wouldn't be enough time for it. Plus, it would kind of get into the political side of things and if the prequels taught us anything, it's that too much politics is not that good in a Star Wars movie.