I believe the end result will be a permanent split similar to BTC/BCH and ETH/ETC, with Bitcoin (BTC) taking the branch with the bigger blocks.
BTC/BCH is not a "split". BCH is a copy coin like hundreds of other copy coins that copied bitcoin's code and made small changes to it.
This is interesting, I haven't heard much talk about larger blocks AND second layer, the conversation has always seemed pretty binary.
What makes you suspect the answer is a combination of the two, and in which order do you think the changes need to occur?
Because nothing can replace the on-chain transactions and people will always want to and need to make them.
Also anything built on top of bitcoin's main chain such as second layer depends on the main chain (first layer) for its transactions. Each time someone wants to open a LN channel they need to make an on-chain transaction. So the on-chain capacity has to be able to handle this new use case as bitcoin and LN adoption grows.
Of course good work has been done to both increase the capacity (SegWit that increased the block size) and reduce the transaction size (upcoming Taproot/Schnorr soft fork) to increase efficiency both contributing to on-chain scaling. But this is still not enough. As Andreas Antonopoulos once said, scaling is not something you solve once. It's a problem we have to always deal with. We improve it a little bit until the future then work on it again and again, advancing with technology advancements and adoption growth.