Note that there were services that ran full nodes and then supplied users with an "account".  Most of these services that existed at that time were either eventually hacked and had their user's bitcoins stolen, or ran off with their user's bitcoins themselves.  Please take note that an "account" is NOT a "wallet" no matter how much Coinbase wants you to believe it is.  With an "account" you are essentially giving your bitcoins away to someone else, and they are promising that they will send a certain amount of "their" bitcoins from their wallet on your behalf.  You have to decide if you trust them to follow through on that promise.