Online wallet is not very trusted in my opinion. I know there is many online wallet which have been running for a long time but their is no guarantee. Not even online wallet every wallet which don't give you private keys or even acces to your wallet cahain is not good. Hardware wallets are most suggested wallet for any kind of holding. If you can't afford bthat use any other wallet wich give fully access to the wallet your key phrase everything. I never used online wallet and I would never be.
Basically, funds should be kept off-line. That's why we have a type of cold wallets. Online wallets are risky as hot wallets. If coins already listed on good exchanges, why not store your coins on those reliable exchanges? It is much better than storing them on online wallets. However, storing coins directly on exchanges is still not a best option, but you can use it as alternative for online wallets. And you should remember to choose reliable exchanges, only.