The blockchain has an annual capacity of around 200 million simple transactions. Thus, if every resident of the USA opened and closed one LN channel (causing three transactions), the blockchain would be full for more than three years! Ergo, there is no need to worry about miners not collecting enough fees.
If adoption was very high, I'm sure exchanges would have implemented being able to withdraw to your payment channel, so those people would not have to keep opening and closing a LN channel. Once Lightning becomes 'mainstream' for Bitcoin users, the only on-chain transactions would be likely:
- Moving to cold storage/withdrawing from cold storage
- Funding physical bitcoins
- Moving a large amount of funds around. Your coffee purchase doesn't need to be immutable, but sending someone a few million may need it
- The occasional closing/reopening of a channel for whatever reason