Hi Steve,
Sorry about your unexpected corrosion!
Rust, or iron corrosion, isn't like fungus, covering it up will stop it from spreading. HOWEVER, simple painting over rust is not ideal, since the iron oxide layer is weak and not bonded well to the steel beneath it. Therefore surface preparation is one of the most important steps in applying a coating to prevent corrosion (something 70-80% of the work in coating steel infrastructure is surface prep).
A metal brush followed by light sanding is probably the best option, but just a metal brush is probably good enough if you can visibly see that all the rust is gone. Chemical removal (like vinegar) is less than ideal, as it is messy and less effective overall.
Finally, I know nail polish is a popular choice for recoating, but nail polish is really just a nitrocellulose lacquer meant for cosmetic purposes and not to inhibit rust. At much better, but still simple and clear coat option is linseed oil, which is the basis of oil paint. It does take much longer to dry, but is much more durable and better coating for inhibiting future rust. applying heat to the area with a heat gun or hair dryer even for a little bit can help speed drying times.
Best of luck. And don't worry too much about it :)
Cheers,
Ben