I figured you were talking about steelies having seen your great posts on resources for getting custom wide wheels. I too have pondered using 13" wheels, in ultra-wide widths. I guess that stems from another Deserter owner (below) who has posted pics of his race buggy with them - just something about that "look" with ultra-wide 13" wheels and matching tires, like an earlier era F1 car. The mind exercise gets difficult trying to source available tires (besides race tires) for these wheels. Slim pickin's for tires, but there are a couple of sources I've come across. Another is the re-produced Michelin TB15, but lots'a $$$ for these!
There can be a lot of reasoning behind tire and rim proportions. Some of which were already mentioned: change gear ratio; change ride height without changing suspension geometry; and style.
There are probably more than I know, but here are some:
-Larger profile tires are good for laying down power with maximum grip. Drag racing tires are made to "wrinkle" on launch.
-Thinner tire profiles are typically more responsive to turning, but they are harder to manage and can yield longer lap times.
-Thicker tire profiles are less responsive and have less driver feedback, but are more forgiving.
-Stepping up the rim diameter increases unsprung weight. Stepping down, decreases it. I do not think that this does as much as people think.
Everything except for the dragging was from an article I read a few years ago in either GRM or Sports Car.
I think that the tire contact patch size is also affected: thinner profile yields a smaller patch while thicker yields a larger patch.
I wish I knew enough to give you advice on specific choices. There is a lot to tire and rim selection, but I would not give much thought to it unless I were at the top of my game.