If we assume that there will also be N hops, and each of the N hops to support the N transactions, then doesn't everything cancel out in the end? Sure, a single hop that handles 10 transactions will replace the 1 on-chain transaction, but that's 1 hop. The other 9 hops also do the same thing, so in the end, 10 transactions require 10 on-chain channels (one channel per hop).

In the mining case, let's say there are 10000 participants in the pool. Then it needs to open 10000 channels. Now if the pool has a ton of BTC up front, maybe it can benefit, but it's pretty painful to create these giant channels for each participant up front. And a participant might disappear, in which case, you pay the closing fee and then maybe open a channel for a different participant. This "churn" of channels can become rapidly expensive.

Now if you say the mining pool doesn't need to open a channel to each participant, now the participants are stuck without inbound liquidity. They can buy it, but that's again, on-chain TXs for each one. And that step to buy liquidity is not exactly easy or cheap.