Two things:

1. You never tell your function to parse a date from the string, and
2. You always return a string from the current date: return formatter.string(from: Date())

Try this instead:

extension String {
    func roundTripDate(style: DateFormatter.Style) -> String? {
        //set things up
        let formatter = DateFormatter()
        formatter.locale = Locale(identifier: "es_AR")
        formatter.timeZone = .current

        formatter.dateFormat = "yyyy-MM-dd" // this is the date format coming in
        //this guards against a String not being in the correct format
        guard let date = formatter.date(from: self) else {
            return nil
        }

        formatter.dateStyle = style // this is the date format going out
        return formatter.string(from: date)
    }
}

let example1 = "2022-09-08"
print(example1.roundTripDate(style: .long))
//Optional("8 de septiembre de 2022")

//string is not a date
let example2 = "Hello world"
print(example2.roundTripDate(style: .long))
//nil

//string is a date but in the wrong input format
let example3 = "12/22/2021"
print(example3.roundTripDate(style: .long))
//nil