What's exaggerated?
This part:
The point is merely that, without some disambiguating phrase or other clear indication near the title, they will not immediately see that they are on the right page,
The very first line in the article (should, according to our guidelines) provides a clear indication. If that isn't immediate, then I don't know what is.
This part too:
and if they know about history, they will naturally wonder, without some clear indicator, whether we know that there are other Charles IIs.
A link to a disambiguation page -- which should be there anyway -- would settle any doubts, but I have trouble imagining that anyone except maybe 15-year-old WP admins would try to seriously suggest that we obviously don't know about the other monarchs that went by such-and-such a name simply because we didn't include parenthetical disambuating information in our titles.
Disambiguating the article for humans the same way we do for the software is (marginally) clearer for the reader and simpler in point of policy (one less thing to explain to people).
I don't know that it is really simpler in terms of policy either. We already spend an awful lot of time explaining how to use disambiguating info.
Sure, we could just expect people to read the first sentence and figure that we know about the other Charleses, and I'm sure we wouldn't go that far wrong, but why deliberately create this problem? Similarly, we could simply ask people to add a disambiguating line just below or above the article title, like "This article is about the French king. See [[Charles II (disambiguation)]] for the other Charles IIs." We'll probably ask that anyway. The point is that the current system of including the disambiguating phrase in the human-readable title is more helpful to humans than not having it. Or so it seems to me. This isn't a huge issue, and maybe I'm just off.
Actually, it's not that big of a deal to me either. I mainly just want to be able to call things by their names. The recent (ongoing?) debates about US versus U.S. versus United States provide a pretty clear example, I think. The disagreement ended up revolving around what things like the U.S. State Department are actually called. Prof. Jensen pointed out that, in that particular case, the department is actually called the "Department of State" (I think that's what he said) and that the name doesn't actually include any reference to the country at all. While I agree that something like [[Joe Quick (King of my own imaginary island)]] is unlikely to actually cause confusion, I think that appending the words "United States" or some variant in the title of such articles actually obscures a minor but important bit of information that many of us who participated in the debate evidently didn't know.