"With that being said, the companies he worked for issued 1099s in his individual name, and not the name of the LLC, which it should have been had his accountant given him the correct advice when mentioning the LLC."

As an Individual with a Single-member LLC, as a Disregarded entity, the customer of your client did it just right. Your client would fill in the W9 as the individual's name, optionally also use the dba, and the owner's SSN or EIN.

"would be much better to have the company in question amend and resubmit the 1099s for 2020"

They didn't make any error. In this scenario, that 1099-NEC is not for the business entity as separate from that owner, in other words. And now, he needs to submit a new W9 if he expects 2021 to be reported properly; well, not reported, by his customer.

"filing the late election,"

Late for 2020 or Late for 2021? They're both late, but if there was no 2020 payroll, it's hard to show they acted "like" a corporation.

"or, if I should include the full amount on the Schedule C, and take a deduction for the amounts that were included in the S-Corp."

It's not Expense. It wasn't paid out for goods and services. You are asking to allocate it to the other entity; not to Expense it.

"depends on contracts and which bank accts etc"

Exactly. Is there a Checking account for that EIN, or is it all running through a "personal business" checking account?

Review 2020. As they say, if it walks like a duck and quacks like a duck, it was a duck, in 2020.