Having a private key (or wallet seed) does not mean that you are safe, a lot of open source/famous wallets have been hacked or lost some keys for example: newbie's wallet (blockchain.info) ----> Old bitcoin history: @johoe return 870 BTC to the owners.
Avoid using untrusted/closed source wallets and it is better to download a full-node wallet.
Downloads only from official websites
If I'm not mistaken, current size of bitcoin blockchain is over 230 GB, and this is main problem for people who want to use full-node wallet. It is not only problem in fact that user need to have hard disk of at least 250 GB, but also in speed of user internet - this is not something for low speed internet connection.
You are mixing online wallets with desktop / hardware wallets, it is not the same level of security and it will never be. Online wallets are most insecure way of keeping cryptocurrency, and by using any desktop / mobile wallet this security is improved. If user want to improve security even more, hardware wallet is next step. Having a private key or seed is just fine&safe, problem may be only in a way how users is handle with such data.