Mere possibility of running a node on your own and sending money directly alters the dynamic. You can use a third party service if you like. You have to use a third party to transfer dollars over the Internet.
I'd say most money lives in banks. Most cash is out in the world.
A downside of having someone else hold onto your BTC on a website is that you can't spend it in the ways that are supposed to make BTC powerful.
On the other hand, you do still reap the benefits of the rock-solid wealth-storage device that is BTC. And where could be a safer place for your BTC than a website on the Internet? Those things never get hacked.