If lightning network is well adopted, it will reduce the on-chain fee dependency to certain extent but may not likely be significantly, many people will still rely on on-chain transactions. Miners are also very important to maintain the networking, keeping it safe from 51% attack. Many people will just not use lightning network because it is not as safe as on-chain transactions. Lightning network is for daily spendings in which only small amount amount of Bitcoin will be enough while on-chain transactions will be required to open and close a channel. But for large amount of Bitcoin holding, holders will not even want to make use of lightning network for larger amount of Bitcoin, while making transaction of high amount of Bitcoin will be through on-chain transactions for security sake. No matter how adopted lightning network is, it will not be good for large amount of Bitcoin.