He's not the one that has to prove a point. Anyone with common sense can see that it's a middle class family. You're acting like it's unrealistic that a middle class family can afford what that family has, when it happens all the time in our world. Payment Plans are a thing that exist, you know? It's a thing people can do where if they want to say, pay for a expensive car, they don't have to pay for it all at once, but can do it in monthly increments. And believe it or not, Middle Class families can afford things like maids and computers, so is it really so much a stretch of the imagination that our protagonist's family can afford a Robot Butler, when such a thing was being heavily advertised and sold all over the place?
Given the era Fallout is based off of, it really is not out of the question that society was based heavily off fulfilling the "American Dream." And anyone who has taken a very basic American history class will know that Post WW2 America which again, the culture of Fallout's pre-war society is HEAVILY based off of, was a time where spending, and capitalism as a whole, were EXTREMELY encouraged all throughout the country.
So, unless you have some kind of secret, firm facts that prove your theory, you're wrong. And if you still want to deny it? Fine. But please, don't try to force your ignorance onto the rest of us.