You can’t do cold storage on the Lightning network. All the $$ that you put into a payment channel will be similar to storing your money in hot wallet and will be vulnerable to theft.
Because of the higher risk of theft, it won’t be smart to keep a large amount of money in a payment channel. But, because of high fees on-chain, the only way to make the opening and closing fees economical is to open a channel with a large dollar amount.
Right now fees are $5 for a normal transaction, which is significantly down from a month ago. At $5/txn ($10 total to open and close) you need to put $1000 onto a payment channel to have it be 1% transaction fees (ignoring the fees that you pay in the LN).
To be compete with BCH transaction fees, you would need to be adding a lot more in USD value to each channel. But having that much money sitting in an LN channel is dangerous. It opens you up to theft just like having a lot of money in a wallet on your phone opens you up to theft.
There are lots of other potential issues with the Lightning Network, but the fact that you will need to store large sums in a less secure environment to make fees comparable to traditional (non-crypto) payments.