"contrary to how the Amended return can be filed if you Efiled the Original, You can Efile the Amended"

That's an Intuit Program provision; not an IRS provision.

The IRS would ask for paper submission when there are things they want included that are not part of what software would typically include as efile.

It seems they rejected your original efiling of that client's return. You can't figure out what more they want, that makes an amended return necessary?