The international influence and monetary income they provide the nation probably have something to do with it.
What international influence?
Honest question, like, what do you think England wasn't able to achieve politically if they didn't have a monarch?
And money?
Are you talking about the Crown Estate?
Also, tradition, which is a less reasonable excuse but likely the most powerful. People enjoy the notion of Kings and Queens - look at the American phenomenon of Renaissance Faires. As an aesthetic, Royalty has a very strong appeal.
And this is what I don't get.
Why do people enjoy that notion of continuing rewarding people just because their ancestors were better than average at grabbing power (which almost always involved killing people)?
And I'm pretty sure if we were paying Renaissance Fair wenches salaries and giving them special legal privileges, people would be pissed, also, I think you're overplaying the popularity of those fairs by quite a bit

.
Secular thought was strong long before the American revolution.
I wouldn't call it long before, but it's true that philosophers like Hobbs and Locke talked about it since the mid 17th century (though Hobbs kinda pussyfooted on the issue and ended up with monarchy), the point I was trying to make is that the American revolution is the first instance where those theories were applied.
Are you American yourself? What you just said betrays a very odd bias and thought context.
I'm not an American but I have been living here for the past decade, and while it's hard to be 100% self critical about those things, I'm pretty damn sure my strong anti-monarchy positioned were formed well before I moved here.
Also, I traveled the world quite a bit, and I don't think that anti-monarchist sentiment are an American thing, far from it.