I don't see how Lightning could solve this at all. If the whole point is to have an immutable and verifiable record then Lightning is the last thing you want to use since by design you make transactions over Lightning without leaving a record on the blockchain. And even if you could use Lightning, it would still require everyone in the country to open a Lightning channel, and so the number of transactions being processed by miners would be just as large (or even larger, if people also want to close their channels afterwards).