Coinbase is insured, though, which means if something happens, funds should be reimbursed. I'd think that trumps the privacy concerns.
But you relay in coinbase, what happens if they close your account or they have a downtime and you want to use the bitcoins? That is not a smart way.
They do not close accounts for no reason. They have to remain legal so they are not considered a money laundering service, they have spent tons to stay legal or become legal.
Most of the closed accounts I think I've read about (and its been a while since one I saw) were people who bought BTC then went straight to localbitcoins trying to sell it for higher, and repeating this action over and over... many times.
As far as downtime you would be able to do nothing. But you will not find a lot of downtime. I remain with my statment it's a good service for a non-technical person. If someone is not capable of safely storing private key then it's a good option.
Obviously some users are much more experienced, and can handle having private key.
I also want to point out that Coinbase is backed by tens of millions in investment dollars. I'm sure their investors wouldn't like it if they started arbitrarily screwing over their members. The investment is a long-term play, not a short one. And they're one of the few that are fully licensed for LEGAL transacting.