I'm using Blockchain.info. I don't know why, but I see it as a safer more convenient way to deposit my money than an offline wallet and I can use the website anywhere I go with ease. It doesn't need to sync.
FTFY
Blockchain.info is by no means a safer wallet than a desktop wallet. It is a web wallet, at the mercy of the Blockchain.info. If they were to shut down, you would lose your Bitcoin. Also, the wallet technically does need to sync, it relies on blockchain.info's node being synced, which it may not necessarily be.
Bitcoin core introduced pruning in v12 for wallet functions, which keeps down the download size but still requires a bit of a sync:
https://bitcoin.org/en/release/v0.12.0#wallet-pruningHaving your own copy of the blockchain is kind of the point, but you could always use a web-wallet, coinbase, etc.
(Just remember to make sure you have the ability to claim your private key, use 2fa, and a strong unique password)
Bitcoin Core with pruning still needs to sync, so it is not ideal for someone who doesn't want to wait. Pruning simply makes the storage requirement irrelevant, it does not affect the sync time.