I have a client for which I am exploring this option going back 2 or 3 years...and I don't think your response makes logical sense...(no offense intended). 

The only way it makes sense is if the tax payer had not filed any returns for his LLC (in this case the 1120s) AND either a) not filed a schedule C on his personal returns leaving out the business entirely for the year(s) in question or b) not filed his personal returns at all for the year(s) in question.  I don't think it makes reasonable sense that the IRS designed a way for only tax payers who were technically delinquent in filing their returns to get relief for failing to make an election while excluding those who tried to do the "then best thing" by filing something and being in compliance.

Plus, how can you possible file a 1040x to show that you intended to be treated as an S-Corp without the help of Doc Brown from "Back to the Future" since doing so (Filing the 1040x) would require you to have the K-1 from the LLC treated as an S-corp.   If you could get away with filing the 1040x, you'd already be standing on home base experiencing the benefit of an S-Corp without the need for approval from the IRS to be treated as an S-Corp.

I'm inclined (for selfish reasons) to interpret the sentence "entity and all shareholders reported their income consistent with an S corporation election" to mean that the shareholder's reported a "reasonable salary" and made the appropriate and timely tax deposits.

....admittedly I may be chasing my own tail here but I'm looking for answers too...