Many people believe that electronic wallets are safe, but in fact, Bitcoin accounts and trading floors are frequently hacked.
As there is no regulatory body that guarantees a Bitcoin number to a person, it is very difficult to retrieve it if lost. If stolen, there is no hope. Such attacks are ongoing.
You are confusing two things:
1. bitcoin wallets
2. exchange wallets
Saving stuff on an exchange wallet is never recommended unless it is for trading. When you are done you should move it to your personal (preferably hardware) wallet as soon as possible.
Personal bitcoin wallets are only as safe as you make your system.