My personal opinion,
You increase onchain transaction capacity
1. To keep the blockchain fees affordable
Affordable is a relative term. If I'm sending the equivalent of a few million dollars, then a few hundred dollars in fees might be "affordable". If I'm trying to buy a piece of candy that's worth about $0.10, then a $0.05 fee is ridiculous. If I've worked hard and saved for decades so I can finally buy that "thing" that I've always wanted, an extra few dollars might be more than I can "afford" at that moment.
2. To keep transaction process timely so you are not waiting days for a transaction to complete
Pay a competitive fee and your transaction will be timely.
Small point, no matter how fast an offchain solution is, if the onchain capacity fails so too will the offchain.
As their are known dangers to offchain , if the onchain does not broadcast withdrawal info in a timely fashion.
So at some point onchain must be improved or even offchain faces failure.
Small point, if everyone in the world were to start to use bitcoin for EVERY transaction they engage in every day, you won't be able to create a block big enough for them all to be confirmed "quickly". Some transactions MUST be off-chain. It simply won't work any other way. The more transactions that are moved off chain, the faster (and cheaper) the remaining on-chain transactions will be (for ANY given block size).
Also using LN only makes sense if you are transacting multiple times between the same people.
The normal way people pay for gas or food or rent, is weekly, daily, or monthly, but not the 100s of times needed to make LN worth it.
Only Banks process thousands to millions of transactions per day between other banks that would really be advantageous to use LN.
I don't think LN works the way you seem to think it works. Perhaps I'm the one misunderstanding it, but shouldn't you be able to send bitcoins to someone over LN that you don't have a direct channel with? I haven't kept up with the developments over the past 18 months or so, but the last time I looked at it there's no need to "transact multiple times between the same people" for LN to work well.
It really surprises me that Exchanges (such as Coinbase and Kraken) don't set up LN and strongly encourage their customers to receive most of their withdrawals in the form of a LN channel. Wouldn't that effectively turn the exchange into a hub? If I want to transact with ANYONE that ALSO has an account at the same exchange, then my transaction would just need 2 hops (one from me to the exchange, and a second from the exchange to the person I'm transacting with). If the big exchanges then set up a few large channels between each other, I could transact with any of the millions of customers that acquire most of their bitcoins at exchanges.
Same thing with mining pools. If each of the largest pools were to set up LN and strongly encourage their miners to all take their withdrawals over LN, then the pools would become yet another hub. The pools can even set up their channels in the blocks that they create (which reduces the "cost" of setting up the channels to the lost opportunity of including some other fee-paying transaction in that block instead). If they try to do it mostly when the blocks aren't full, then they can effectively open and/or close their own channels for free.