This is a common misconception about Obito's character throughout the Naruto Fandom and beyond.
Obito didn't start the 4th Great Ninja War or strive to complete the Eye of the Moon Plan because Rin was killed, but because Rin being killed was endemic of a larger problem and that was the Shinobi World itself. From the very start, we see just how ruthless and cutthroat the Shinobi World can be, the fact that the Chuunin Exams held by the Hidden Leaf Village, the tamest of all the Elemental Nations, still had numerous Shinobi die shows just how twisted the world of Naruto can be.
And Obito isn't the only person to come to this sort of conclusion, Nagato did as well. He saw how corrupt the world was and sought to enforce his own version of peace where there would be one, over-arching figure-head so powerful, it would deter anyone from starting a war in fear of being wiped off the face of the planet. Obito wanted to create a world where everyone could live out their ideal fantasies without having to shed blood. And of course, for Obito, his ideal fantasy was being with Rin because, honestly, she was the first person to ever show him any sort of real kindness outside of a handful of people. Not to mention, Obito, much like Naruto, tended to get attached to his teammates, and would do anything to keep them alive/happy.