The glue plugs the hole. The rubber is a strain relief so the hole does not pull open when inflated.
In other words, don’t worry too much about the patch. I like Rustines, sold by Velo Orange, because they are much easier to handle than the type with feathered edges. I especially hate trying to get the clear layer off of some patches.

3-5 minutes is about right before applying the patch. If it takes longer, you are applying too much glue. The bond is pretty much instant, but not very strong. It doesn’t need to be.