LN closing transactions are currently designed such that they can be broadcast at an arbitrary time in the future.
This is not true, because they are based on some fee rate, and you have no guarantee that fees will be on that level in the future. So, if you assume that minimal fees will be lower than today, then you are safe. But if you assume they will be higher than today, well, then your transaction could be rejected as non-standard, or could require RBF/CPFP to be confirmed.
Further, if coin is removed from (a subset of) the UTXO set at intervals less frequent than every block, there will be incentives to get transactions confirmed prior to this frequency, and as such, the cost of getting transactions confirmed will spike immediately prior to these block heights.
We already have those spikes, they are called halvings. So, the solution is to make fees more smooth globally, by adjusting coinbase transactions, not by adjusting users' transactions.
increasing the total coin supply is the simplest way to achieve this
It is not the simplest way, because:
1) if it is a hard-fork, then it can be rejected by just doing nothing (I won't upgrade to the tail supply version, what then?)
2) if it is a soft-fork, then miners could say "no", and people could burn or lock coins, to resist "tail supply attack", and remain in the same network (I prefer locking, because it is more resistant to this attack, and people can always agree to spend coins, that are circulating in some loops, to keep those coins away from "the legacy supply"; expect every additional satoshi to be tracked and blacklisted, and to be used to counter-attack)
3) if it is a no-fork, then it has the highest chances to be introduced, and it is unstoppable at the same time, because you can keep it on the sidechain; then if tail supply enthusiasts are right, users will join them (I expect this chain would fail, or would need being "rescued" by burning all additional coins)