Yes, but the "numerous transactions" wouldn't go in the miners' pockets if LN activates.  So they would have a small fraction of the "numerous" transactions at low cost.   You clearly see from the fee revenue of miners, that scarcity is more efficient in rising income, than high volume.  You may think that in the long run, this kills bitcoin.  I'm not sure, because a reserve currency is exactly having these properties.  So this niche is what is open for bitcoin: reserve currency, with very few transactions, at very high amounts, between big institutional players using it to settle their unregulated affairs.
These big settlers don't mind paying large fees, because compared to their business, it is still a small cost.  
Bitcoin is not made for the masses ; like gold is not made for the masses.
The masses just served to get bitcoin up and running but at a certain point, the masses will become a nuisance.