Why is Destiny re-installing!? Why would it have to install in the first place if you bought the disc!? This is ridiculous! Bungie, you are the new Comcast. Do the world a favor and kill yourselves. But not before handing this game over to a competent game company that could run it with no issues.
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Sounds like you're a little late to the "giant update" party. Everybody else already complained about this. But all games install from the disc these days. Otherwise you'd be sitting there for ages at every loading screen. That's why consoles have a hard drive. I'm surprised that you haven't noticed this on every single other game you've played, unless Destiny is your first video game since 1991.