I read this question and instinctively agreed with Agent86's answer. But I decided to dust off my own SNES to do science and confirm his answer. The results were not exactly what I expected.
I took out my North American SNES and my library of 13 games. All 13 (eventually) worked. I plugged in one of my controllers in the Player 2 port and tested them out. Results:
Out of the 13 games, 5 recognized and worked with a controller plugged only in the P2 port.
5 out 5 of those games supported two player mode.
4 out 5 of those games allowed you to start a single player game with a single controller plugged in the P2 port. The other game would only let you choose two player mode.
The other 8 games were all one-player games only.
What I am concluding from this experiment is that you can play some SNES games with only the P2 port functioning, but it seems those games have to support two player gameplay with a second controller. Some of those games have good one player campaigns (Super Mario World, for example), but solely one-player games seem to be a no-no.