The so-called fee market in BTC is not a market, nor anything like that. In a real market, when demand increases relative to supply the price also goes up. But the suppliers immediately react to the incentive that this price increase implies, by producing or importing more of the good and making it available to the additional demand. This again puts downward pressure on prices and makes the good affordable again in the shortest possible time.
However, BTC fees “market” does not work that way. There is a maximum limit to the amount of the good available per unit of time. That means that no one can increase supply even if the price goes up. Bidders compete for a good whose supply is artificially limited, and raising the price does not solve the problem. In other words, those who are able or willing to pay higher rates obtain the good at the cost of driving out others.
Then the surplus demand can only react in these specific ways:
1) By paying an even higher fee and contributing to the price increase.
2) By delaying the transaction until the fees are reduced.
3) By making the transaction with a different asset.
If LN were an acceptable solution for the market, every time we see fees increase there would be an increase in LN transactions, the BTC network would be unburdened and fees would be immediately reduced. However this does not happen, which tells us that the market does not see the service provided by LN as being able to replace that provided by the BTC blockchain. The fees are reduced again only after some time, to the extent that fewer transactions come in than are mined, simply.
So, the most sensible solution is to use a different asset with lower transaction costs, like Bitcoin Cash, replacing BTC.