Basically it will be too much bandwidth for most users to bother running their own core, meaning increased centralization for BCH and easy way to kill it in the future through some government action or bug.
In reality, you would have to be insane to store bitcoins with a C++ program permanently connected to and broadcasting your IP address to a p2p network.
Now, it may be that many bitcoin nodes are run by insane people who do exactly that, but if the foundation of Bitcoin's security is that its users have no understanding of computer security, then the whole thing is doomed to failure anyway.
You run the node and connect your light wallet to it, so that you don't need to trust someone else's node to tell you what is going on. I run mine on an old netbook.
Some people might store their private keys on the nodes themselves, they shouldn't.