So I guess Litecoin cuts it pretty well as a Bitcoin-like cryptocurrency
You can argue centralization problems, but yes.
If so, how can you then claim that it will have the same problems? As far as I understand it, payment channels are created to specifically address the issue you are talking about (i.e. running out of processing capacity). Therefore, it is not about me arguing that payment channels can have infinite processing capacity. This is basically what they are there for exactly. In this way, your whole claim is not valid
That's the thing. You don't understand it. Three assumptions are made, and all of which are implicitly understood by those who understand the model of LN. They are:
1) A single payment channel is enough for you to transact with everyone (routing).
2) Everyone that you want to transact with is on LN.
3) Everyone has already opened a payment channel.
Assuming 10 million users (random metric), calculate how much block space it would require for each user to open a payment channel. Please post your results.
Anyhow, it isn't exactly recommended for high value transactions at this time.