If you remain online, and if the block chain is not saturated, however, the LN is quite safe to use.
Would it be more viable then for the Lightning Network to be more like a web service than something you use as a desktop application?
Only if you give your keys also to that web service. In other words, your bank. You have to stay online to be able to sign at any moment with your own keys. If you want another service to do so while you are offline, you have to give them your keys. But then, they have your coins. Like an exchange, or a bank.
You're failing to mention another option (that may well be the most popular option):
Set up a uni-directional channel. Then you keep your keys, and go offline whenever you like. You're turning into another one of those people that keeps saying the word "bank" as often as it can be fitted in.