The design intent of HNQ is to provide interesting reading. The HNQ draws from 170+ sites, so the odds that any given reader will have the expertise to answer a random hot question are very low.
But you're describing a case where the HNQ leads you to a question you can answer, because while you're not an expert on the site's overall topic, you have specific knowledge here. And when you get there you can't answer because it's protected. That's frustrating; I've been there too.
HNQ isn't the only path that leads there, though. I see questions that have been tweeted, questions that are linked from other questions (on other sites), and questions that hit Reddit or other sites. And if those questions are protected I can't answer them either.
The HNQ is global; there's no per-user filtering (and it would be expensive because of how often the list changes). The only way to solve your problem would be to bar protected questions from HNQ entirely. That wouldn't solve the Twitter/Reddit/etc problem, but at least it would stop you from having a bad HNQ experience. And I should also mention that, as a moderator on a site that often has stuff on HNQ that we'd rather not, I could get behind a "protection blocks HNQ" idea.
Except that the questions that get protected are usually the most popular and active, the ones that -- for better or worse -- will actually draw visitors to the site in question. I'd like to see changes to HNQ management, including a way (short of closure) for a community to remove a specific question from it, but this proposal seems too heavy-handed. It's also likely to produce a flurry of anticipatory protections across the network, making it harder for visitors to contribute and help grow our sites and communities. Maybe that's fine on big sites, but the smaller sites I'm on don't want to pre-emptively drive prospective users away.