Well, except that it wouldn’t be writing its laws based on aligning them with religious interests, which is pretty much the very definition of a theocracy. So other than the minor issue that it’s the complete opposite, spot on.
No, the OP hypothesizes a country that had values that would put it in line with the US constitution except for this one thing. That’s very much not China. We’re talking about a culture/governmental system with one tweak, whereas China is about as alien as you get to the US way of governing.
My opinions on this issue are conflicted. If we were talking about a revolution in an existing country that made these changes, I’m not comfortable enforcing this idea onto an existing population. However, if a piece of land were the suddenly pop out of the pacific ocean and for some reason we decided that it was going to be basically the US + this minor tweak, and everyone knew going in what they were in for, then I’d live there. It’d probably be the best country on earth - it would self-select for people who were on average much smarter than the average person.
This is one of those tricky issues where you have to balance the ideal of freedom against the practical reality of freedom. For instance - when neonazis want to give speeches in the middle of town square, nothing good is going to come from that. It does nothing but drag down our society. But it would be ideologically inconsistent for us to prohibit this, because even though it’s a use of freedom to make society a worse place, some people think a consistent ideology of freedom is more important. Or, say, letting really stupid, uneducated people vote - this undoubtedly has a negative effect on the state of the country and probably ultimately our freedoms, and yet, can we deny these people the right to vote in the name of protecting our freedom? It’s ideologically inconsistent.
So, here, too - religious people in office - I mean, the truly religious ones - are a net negative effect on society. They also tend to be harmful to freedom in a lot of ways. So by weeding them out, we’d probably have a net increase in freedom - and yet it’s ideologically inconsistent because we’re doing this at the expense of not allowing certain people to participate in government.
On these types of issues, I tend to side with the side of ideological consistency and against pragmatism. But I’m not so sure it’s clear cut. So I think that this hypothetical society would be a better place to live, yet, we might feel bound to work against it and instead enable policies that allow it to become worse, in the name of freedom.