There are multiple reasons why this is a very poor idea.
For a start, minnows (and sticklebacks) are very sensitive to heat and the minnows need very high levels of oxygen. They are no more 'hardy' when it comes to ammonia or nitrate than any other fish, and considerably less so than tank bred fishes. The waters they live in might look filthy and poor, but in terms of toxins, they're generally quite clean; zero ammonia and nitrite at all times and very low nitrate, if any at all.
Cycles go faster at higher temperatures, so you'd be along time cycling in the cool waters that hose fish prefer.
Wild fish could carry pathogens or parasite that they are immune to, but that aquarium fish would not be. Anything in your tank, like decor or plants, could also infect the wild fish in the same way, so you would never be able to release them back into the wild, you'd have to keep them permanently.
It's also illegal in many places to catch wild fish; although I doubt you'd get into trouble over a few minnows, but that's hardly the point.
Anyway, the reason fishless cycles were 'invented' was to save any fish from being harmed; why would you want to inflict potentially lethal amounts of ammonia and nitrite on any fish, even wild caught minnows?
As to why some fish are hardier than others, it's all down to where they evolved to live. If there are rivers or lakes that experience ammonia, then I'm sure some fish would evolve the ability to cope with higher levels.