What rhetorical gain is expected by saying “Do they really not understand why?” instead of “Don’t they really understand why?”

Those two sentences emphasize different things, which means that in a sense they're asking different things.

Don't the country's leaders really understand why that won't work?

This puts the emphasis on understand, and is thus a straightforward question: do they or don't they understand? (Also, in this version, "really" modifies "understand", which in my mind kind of weakens the question by giving the leaders an "out": we assume they understand a little bit, and the only thing in doubt is whether they understand the issue well.)

Do the country's leaders really not understand why that won't work?

This phrasing puts the emphasis on really, and thus instead of being a straightforward question, it becomes an expression of incredulity. We're not really asking whether they understand or not, because all the evidence says they don't; instead, we're asking whether we really ought to believe that evidence. (In this version, "really" modifies "not", which is what makes this into an incredulous questioning of our own sanity, almost, rather than a query about what the leaders know or don't know.)

It’s making really not stand out stronger.

If you speak the sentence aloud, those both take stronger (and probably equal) stresses that way; they aren’t reduced and hurried through the way happens with contractions.

Try “leaning” on them really hard when you say it aloud and you’ll see.

There are some distinctions between the forms.

"Don't they really understand...?"

This form is more matter-of-fact and assertive in it's stance. The writer/speaker is implying that they actually do (or should) understand this, which wouldn't necessarily fit the intent of the writer in this case.

"Do they really not understand...?"

This form can be more ambiguous (and may be done deliberately to shield the writer). Taken at face value, and with an even emphasis on the words, this can be an honest question: "Is it the case that the leaders do not really understand this?"

However, as mentioned in the answer from @tchrist, "really not" is often interpreted by the reader as having special emphasis. This is often used in a rhetorical sense. The writer could be interpreted as belittling their lack of understanding (the phrasing in the quoted source is almost conveying a sense of the matter being obvious), or even call into question the veracity of the associated statement. Many readers may apply that emphasis (as I also did) - and the writer could very well be counting on this, from the relative safety of the ambiguity.

Just to give another view into these different forms, consider the following scenario:

Billy, while complaining to his mother about having to go to bed by 9pm, states that his friends next door get to stay up until 10pm.

Later, Billy's mother discusses this with her neighbor:

"Don't they really go to bed at 9pm?"

This conveys a sense that the mother knows/believes this to be the case, and is simply confirming this. This would be akin to "They go to bed at 9pm, right?" While there is still room for uncertainty on the part of the speaker/writer, they are still asserting the statement that follows, and will believe it unless corrected by the other party.

"Do they really not go to bed at 9pm?"

Depending on the emphasis, this statement could have a number of meanings.

a) Is it the case that your children don't go to bed at 9pm? (honest question)
b) I'm pretty sure my Billy is pulling my leg and angling for a later bedtime. Can you just confirm that your children also have a 9pm bedtime?
c) I can't believe you let your children stay up past 9pm? Please tell me I'm mistaken.
d) What kind of parent allows 6-year-old children to stay up past 9pm?

While these distinctions are generally conveyed more clearly through spoken emphasis, they are relegated to the land of ambiguity in written form, unless specifically formatted. Thus, the writer may be able to shield themselves from (or at least deflect a portion of) the wrath of those mentioned, while most savvy readers would likely pick up on the rhetorical nature of the statement.

In British English at least, "I don't really understand X" is often a rather weak statement about something, meaning "I partially understand X but I don't fully understand it", or "X is not very important to me, so I don't spend much time thinking about it", or "I know how to use X successfully, but I don't know why or how it works". "I really don't understand X" is stronger, meaning "I don't understand X at all, and I know that I should understand it".

For example many people might say "I don't really understand the Internet" (i.e., "I use it all the time but I don't know or care how it works"), but if a mathematics undergraduate says "I really don't understand calculus" that is a more serious problem!

There is the same difference of emphasis between the OP's rhetorical questions. "They don't really understand" becomes "Don't they really understand...?" "They really do not understand" becomes "Do they really not understand...?"

"Don’t the country’s leaders really understand why" is incredibly clumsy, and I wouldn't know at all what exactly the speaker was trying to say. (I could figure out what the sentence means, but that isn't necessarily what the speaker tried to say).

"Do they really not understand why" is absolutely clear. It is a rhetorical question, not a question. He expresses his opinion that "they do not understand why". The "really" expresses the high incredulity of the speaker; in other words they really should understand why, and it is incredible to the speaker that they don't.

"Don't they understand why" would be a "normal" rhetorical question, without the added incredulity. "Do they not understand why" would have an accusational tone.