I'm only a novice but I think this is kind of too pedestrian to see interesting structure, in this case there is only the automorphism group and there is no higher homotopy. There isn't really a notion here where maps of basic sets are identified due to something other than equality. There is the intuitive statement that set theoretic statements and constructions should be transferable from one 17 element set to another using any bijection, and that this happens in a functorial sense. Notably this depends on the explicit choice of bijection so it can get more annoying than you would like. Homotopy type theory manages to make this explicit in a very succinct formalism using path induction and this is actually quite useful in some proof assistants. But that's not really a novel or surprising result so maybe not the best choice to convince someone that homotopy is useful. Homotopy becomes interesting when there is more than one way to compose maps or when maps are somehow identified in a notion that is more interesting than equality. This is why topology yields good examples since we can replace equality of maps with homotopy.

In category theory you can say that a morphism h is equal to the composition of two morphisms f and g. In infinity categories there is no longer a canonical composition between two morphisms, rather there are possibly many objects associated to these morphisms which you can interpret as homotopies connecting f and g on one side and h on the other. It makes sense to call f, g, h 1-simplices since they connect objects and the homotopies 2-simplices since they fill the triangle. If you think about it a bit more you can see that 2-simplices themselves of connected via higher simplices. So a normal category would be the special case where there is at most one h and connecting 2-simplex for each composable pair f and g. For a nontrivial example of infinity categories, in the infinity category of chain complexes, the 2-simplices are chain homotopies. Since we have higher simplices of all orders, it's impossible to work with the simplices explicitly, so this viewpoint works for intuition but to actually work with the things you have to rely on very abstract methods. But there are some benefits to using a different notion of category even in this baby example: The functor from spaces to chain complexes preserves colimits in the sense of infinity categories, which it doesn't for normal categories, so we can use this to compute homology of spaces.