What happens if your I-20 runs out while you are waiting for your checkride (in the USA)?

I think the first option couldnt apply to me as I finished my training and was signed off and my I20 runs out very soon , the important thing is that the i20 is still valid in this case. Also the visa would have to still be valid so that I could reenter the US .
I think the second option makes more sense in my case , as my visa has expired too , I wont risk going back on a holiday visit. Its getting clearer thanks Peter .

Pardon the possibly ignorant question, but would this shortage also affect getting a flight review for a 61.75 based piggyback?

No, any instructor can do a flight review, the shortage is about examiners