Dragonball Z - Rule for dragonballs
You made a small mistake, with regard to the rules of the earth's Dragon Balls, they can be used to grant up to 3 wishes. However, if one of the wishes involves reviving a large number of people, it can only be used to grant up to 2 wishes.With regard to your main question, as to whether the Dragon Balls would work on another planet, the answer is Yes, at least in case of the Namekian Dragon Balls. We know this because they were used to revive Krillin, Yamcha, Tien and Chiaotzu on earth which you can see here.Hence, to answer your wife's question. Yes! It would've been certainly possible. However, there are reasons why that wouldn't work with the narrativeThe first is Frieza's character. Frieza was considered to one of the strongest fighter in the Universe. A Universal emperor and it would be very unlike him to take the dragon balls and hide it when he could take it from anyone he pleases simply because of how powerful he was. The Frieza then didn't fear anyone. Although the Frieza in DBS would do something like that, the Frieza in DBZ wouldn't simply because of how strong he was.Now with regard to Vegeta, he began collecting the Namekian Dragon Balls when he heard about Frieza wanting to collect them. So ideally, Vegeta would want to collect the Dragon Balls immediately before Frieza does and make his wish. At the same time, Vegeta had absolutely no intention of confronting Frieza or engage in combat with him. If Vegeta spent time taking every single Dragon Ball he had to another planet, came back and repeated the process, Frieza would've most likely found a Dragon Ball  during this period of time. This would require Vegeta to confront Frieza and engage in battle with him which is something he clearly wanted to avoid at that point in time. So on account of the situation, it would have been impractical for Vegeta to do this fast enough to be able to get all Dragon balls and wish for immortality.