Maybe pure online wallets are by far more convenient to use?
Exchangers got somewhat tight regarding 2FA, cold store, withdrawal limits, KYC, AML and what's not. But!
Makes perfecly sense storing coins on a tradeside. There is a reason for using tradesites.
The opposite using a "Web Wallet" makes absolutely NO SENSE. If you do not own the privkeys, you do not own those coins, period. A "Web Wallet" can do absolutely nothing besides stealing your coins.

Try a light client. People who cannot download the whole BTC blockchain (soon there are two blockchains to download) use those quite often.