Yes, patience is often useful. With enough of it, good hand-eye coordination and either a round Swiss-pattern file or a Dremel tool, you could have opened up the hole to the head diameter of whatever screw you first acquired--without needing to know the actual diameter!

Or you could have used that tapered repair reamer to very precisely open out the hole. In metal that thin, the difference between a cylindrical hole and one that tapers slightly would be meaningless.

(Mantra: it's good to know a way of doing whatever is needed, but it is much better to know five different ways, because then you can choose the best one for the situation at hand.)