I'd like to add that I don't believe that Lightning Network is the best solution to scalability. While it can certainly address the currently high fees, I have some long-term concerns about relying on it. The thing is, the Bitcoin network was designed such that less coins are minted over time so that the miners can gradually transition to being paid in fees. Eventually the only financial incentive miners will have is transaction fees, which means more transactions would lead to better mining incentives.
Using the lightning network to open channels through which several transactions can flow without fees circumvents that and can actually lead to higher average fees for everyone else. At present, it would work great because that would remove from the Bitcoin network large amount of transactions that would otherwise congest the network and improves usability for smaller transactions, but that doesn't side step the need to address network scalability issues. If anything, it seems like the lightning network would be better used as a tool for decentralized exchange between cryptocurrencies, but shouldn't replace the ability of any particular coin to act as an exchange of value.
If node operators were rewarded by the network the way miners currently are according to their bandwidth they would have a financial incentive to scale up their capabilities leading to the network as a whole scaling up if dynamic scaling was implemented. That would ensure that transactions everywhere confirm cheaply, and the lightning network can still be of use for quicker in person transactions at stores where you don't want to wait more than a few moments for the transaction to confirm.
If we go the route of relying on secondary networks to exchange value, how exactly are the miners going to be paid when there are no coins to mint and no on block transactions to confirm?
To answer the first bolded text: While you are not wrong about the nature of decreasing the block reward in order to put the focus on the transaction fees as the rewards for miners, the scaling issue of transaction fees does not really relate to that. What matters is where the reward lies, not the amount. The amount will always be related to the supply and demand of the available hardware in the network. If transaction fees decrease, miners get paid less and therefore mine less, which lowers the overall difficulty of mining and makes it more profitable for the remaining miners who are still in it. If they increase, the number of miners will increase with it as will the difficulty to balance everything out.
To answer the second bolded text: There will always be coins to mine. The Lightning Network still depends on miners. All it does is perform the reconciliation of may more separate transactions off-chain into a single final entry that once "closed" gets put into the blockchain. To get put into the blockchain, miners still have to choose the transaction and will still get paid a fee. The difference is that the fee will be much less than it would be with all the separate transactions, which is fine considering the supply (miners) numbers will adjust accordingly.