No, this isn't an argument for big blocks, I'm just saying how it is. As for Lightning Network, I imagine it would also have a lot of routing failures if the amount of transactions is past some tipping point - no system is infinitely scalable.
Big blocks would also suffer from the same transaction overload that the OP's payment processor experienced, except instead of hard failures, there will also be very slow transactions.
In the end, it was only a short blip. Probably less than 30 minutes of disruption. 20 minutes of very slow transactions and 10 minutes of failures. Still annoying to deal with, though.
I'm not impressed that at this point, with Black Friday + Covid19 traffic, there are people who still think that VISA's transaction speeds are the holy grail for bitcoin, since an error like this can happen to anyone. There may one day be an even bigger load that overwhelms their transaction throughput. For the back-end folks, throughput limitations are a fact of life.
Sure, they can always scale, but you can only scale so much vertically using optimization techniques. After that you can only scale horizontally by buying more processing power but you will eventually hit a budget wall.