Interesting points. I think you are both correct, and I was somewhat correct. I found the current collective bargaining agreement for the NFL. Article 6, section 2, paragraph d of the current NFL CBA states "No player shall be eligible to be employed by an NFL Club until he has been eligible for selection in an NFL Draft."
Other sections of the CBA lead me to believe that everybody has a draft-eligible year, whether they even know about it or not. Your draft year comes when you are three years removed from your high school graduation, and you are no longer eligible to play NCAA sports. It seems that players needs to apply for special draft eligibility to forgo remaining college eligibility. I believe as an engineering student who took 5 years to graduate, I had a draft eligible year in 2004. So like Gable, I could now sign with an NFL team as a free agent (ignoring the absurdity of and NFL team singing a middle aged engineer).
I think Gable, the Australian kickers, and the Rugby players probably fall into the category of "they already had a year where they would have been eligible for selection in an NFL draft."
Of course, I am not a lawyer so take this analysis of the NFL CBA for what it is worth, which is pretty much nothing.