I'm confused. You say that you don't store bitcoins on Coinbase, but you do interact with them (not clear why you would have to if not storing coins there). Also, if you didn't respond to them, then how do you know some kind of weird activity isn't happening without your knowledge using your account?
I'm not trying to defend them really, but just want to make things more clear. If you really weren't doing a lot of transactions through Coinbase, then why the email?
Coinbase is definitely more like a bank than anything else, but is usefully for purchasing some bitcoins and then transferring those to a regular wallet, and they are getting some large merchants to accept bitcoin, so that is very good for the community