Well, I wouldn’t want to draw in the extra “unfairly” label. I didn’t think Nathan’s stereotype was particularly unfair, as stereotypes go. You see a lot worse in these forums all the time.
And, short answer, no it does not mean that. Long answer, any fair discussion of any group’s beliefs or actions will include recognition of absolutes and variables. Stereotypes help misrepresent variables as absolutes, usually for ill.
Example 1: You cannot be a member of the Christian church if you do not believe in the reality of Jesus Christ. You can say you are, but you’re not. It’s an absolute laid down by Jesus himself (and denied by clergy who believe in the theological legitimacy of excommunication) that only God gets to determine who is a Christian, and he says “whosever believeth in me” etc. Saying Christians believe in Jesus is not stereotyping, it’s accurate.
Example 2: The Republican Party is not anti-gay. You can say it is, but it isn’t. Opposition to homosexuality from within the party is a variable that is constantly changing over time and differs from person to person. Ronald Reagan and Bob Dole worked in concert with the Log Cabin Republicans (which is an equality group within the party, mostly composed of openly gay Republicans). However, many Republican candidates (probably most) are anti-gay, and are constantly loudly spouting off about it, and claiming all Republicans in favor of gay rights are RINOs - so the stereotypical Republican is anti-gay.
Stereotypes act to misrepresent variables as absolutes. If you don’t want to mislead the uninformed (such as your children, or clueless Britishers) you need to avoid them and/or acknowledge them. Nathan’s children may very well make bad decisions and be unfair to people simply because they’ve believed their father’s false categorization. That would be sad for everyone involved, and unnecessary, and it’s why people don’t like stereotyping.
But I work with logic professionally, and have worked with types (it’s one of my hobbies) in the natural science sense, so in this debate I will almost certainly be excessively precise and technical. You know the Aristotelian square of opposition in syllogistic logic? Some of, all of, none of, etc? If all fish live in the sea and all herring are fish then all herring live in the sea? Fair warning, I’m more likely to be technically correct than great fun at parties.