Never played the game, but I can't imagine that I wouldn't care to know if I had one or two downs left.
Why? Anytime you're potentially in 4-down territory, what you do on 3rd changes.
Imagine: It is 3rd and 8. You have your dump off route open for a sure 6 yards, and both safeties baring down on him. His chances to pickup the additional 2 yards is best described as "approaching zero." Not impossible, but damn close. All other receivers are well covered and are in varying degree of "unlikely to complete the catch."
If the situation is "make the first down or punt," the dump off is a worthless throw. Indistinguishable from throwing a hail mail into triple coverage.
If the situation is "you've got another down," you take the dump off. 2 yards is easier to pickup than 8.
In Rodgers' case, if he is told "Get 1/2 of it or more and we're going for it on 4th," he potentially scrambles on that 3rd down throw. Even if he doesn't make it, I could see 5 or so yards. 4th and 3 is a lot more tolerable than 4th and 8.
EDIT: Whoops, sorry Half. I think we agree