I'll grant you that it showed poor judgment if he really thought he could stay anonymous. That said, I disagree with basically everything else. I think you're reading
way too much into that.
Where you work will have an impact on your career (I think that's obvious). Part of that is simply the social proof of which company. It helped my career by working for various YC companies, for instance. Were they necessarily better than non-YC companies? No, of course not, but people put stock in it anyway.
The first company you work for may very well have an outsized impact on your career compared to subsequent positions. You should make that decision carefully.
Most importantly, when I go to work somewhere, they shouldn't expect that I will work there the rest of my life. Staying at a company 2-5 years is pretty normal. That's a short enough timespan that you should be thinking about how the company will impact your career after your time with them is done. Furthermore, part of the behavior of a good manager and company is to encourage and facilitate your career growth, even though they realize that doing so may very well cause you to grow out of your position at that company.
He's also a kid, looking at his first career out of college. Did you know the right things to look for immediately and inherently? Do you now? Did some of the things you looked for then end up getting discounted as you got more experience? Do you maybe think you shouldn't hold it against him that maybe he also doesn't know everything?
So no, I wouldn't say that that kid was wrong for thinking about several different aspects of the opportunities he was presented, name-recognition amongst them, even if I would contend that he should maybe put less of an emphasis on it, but that's just the thing he might have heard when he asked the question. Nor do I think that he's more likely to up and leave the company when the going gets rough.