If you want to be in full control of your coins there is no other way then to be in possession of your private keys. In case that you see transaction confirmed on blockchain and address is correct, problem is in your wallet which is probably out of sync, corrupted or outdated. This is something every user can solve, no need to wait for any customer support.

Coinbase is just crypto exchange, and is not perfect as well as others exchanges. I think that in this case OP need to be patient and wait for a response of customer support. Electrum or any other desktop / mobile wallet has its drawbacks, and before using them everyone should study well how they work, and how to use them properly and safely.