Then I guess that there will exist on-chain transactions after all.
Me too ... however, if Bitcoin is very popular (>100 million users) and there is no block size increase at all, on-chain transactions will be expensive. For me, sidechains are an interesting "intermediate" layer between on-chain tx and Lightning. They are more trustless than LN but potentially more expensive, as all sidechain validators will also have to store the transaction history, and so they will try to charge a fee, too.
If we need sidechains or LN+on-chain is enough, in my opinion depends on the final user base. If Bitcoin is a "large niche" of a couple of hundreds of millions users, LN+on-chain (with channel factories) may be enough, while in a "world currency" with 1.000+ million users, some third layer (like sidechains) would be needed. For this case, my preferred model would be a network of inter-connected regional sidechains for operations with stores (brick-and-mortar or online) within the same region.
And according to your explanation, it seems that miners would still profit from the LN via the use of Channel Factories.
I mentioned channel factories as an example how miners could use their liquidity in BTC to get additional fees. However, they could also operate as "standard" Lightning nodes.
What's interesting of the Lightning Network is that it enables atomic swaps which could eliminate the need for an exchange (either centralized or decentralized) to swap from one coin to another.
Yes. These swaps could also be the technology that ends making sidechains more usable, as a "sidechain-to-sidechain hop" could be instant (at least if no on-chain settlement is needed).