The only problem with the LN is that there is still the need to close the channel with a fee on the main blockchain unless we either raise the block limits bit or implement -snip-
The block size has been increased with Segwit already, which makes sure that at least initially, we can handle the increased on-chain traffic. If the entire network starts conducting Segwit transactions, and thus drops the legacy format, Bitcoin's throughput will increase far beyond what is currently possible. Segwit while it comes with various important updates, it is actually a first step towards LN. In other words, we don't need a block size increase, but everyone to start utilizing Segwit to the full.
Also, the more economical activity is happening on LN, the less incentive there is for people to close their channels. I can see a time where people don't even want on-chain transactions anymore. They'll buy and sell Bitcoin through LN.