Yes, not really knowing what I'm doing, I was afraid of breaking something using too much force. Fortunately, (with your help) I realized my mistake was that I was trying to pry the connector off of the board, when all I needed to do was pull the ribbon cable out of the connector. So I finally was able to swap out the controller ports/board with one from another SNES.It's been forever since I took apart my SNES so I can't remember, but a quick googling seems to suggest that all you have to do is tug on the cable (on the motherboard's end) and it should come out.
I really hate when I get to that sort of point where you're reading instructions and taking something apart, and then something stubborn refuses to cooperate. You end up wondering if maybe you need to apply more force, or if there's a trick to it that the instructions forgot to mention (or maybe you just read them wrong), and more force would break everything.
Unfortunately, the problem still occurs, so I'm thinking the issue may require a more complex solution. I might be inclined to just return it and get store credit =(
EDIT: I ended up returning it yesterday.