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I thought the first episode was quite inventive, and quite smart ; I'm quite fond of SHAFT in general so I may well be quite biased. However, the way they paved their way back to the original story using the "amnesia" cliché made me twitch in anger (the the last episodes didn't interest me much.)
There are other shows that strayed from their original material ; I for one liked the original Fullmetal Alchimist anime, because they really tried to give consistency to their story - and I found their answer to the whole Lavoisier postulate quite interesting (explored in the OVA if memory serves ?)
But then again it also depends on your level of nostalgia toward the old material.

Regarding the original topic, I'll have to disagree on Mina being able to do without Akira as a Queen, especially the way it is depicted in the anime. Many scenes, most notably the flashback, insist on her emotional dependance to Akira ; can you really say a Queen that would endanger her kingdom in a childish "my boyfriend's the best !" bet is independant ?
It also rings true for Sanzenin Nagi ; both are powerful characters in their own right, like you state, but they lack a core strenght of anime characters : whether you call it emotional trust, friendship, love, many - if not all - anime insist that a hero can not exist alone. In that way, Hayate is much more than contingent male support ; he fulfills Nagi as a character.
Many of those overly powerful princess-like characters have such Achille heels, and to put it in your words that would be a characteristic of their "moe" : their downfall is ultimately the most irrational yet human-like trait, the need for comfort / love / etc. That's also the whole point of the tsundere or the megane or even the osanainajimi : they need the shujinkou and struggle to admit / fulfil that need ; the klutz / airhead I never understood so...
There is an on-going anime that poses further interrogations on this subject : Arakawa Under the Bridge. Nino appears as a very independant, if not "stand alone", character, yet she asks Kou to be her boyfriend, all the while distorting the usual take on the classic "romance with a princess" situation. It is also interesting in a mirror perspective, as Kou, a gifted and promising young man, ends up living under the bridge because he "cannot depend on anyone".

Regarding the "princess embrace", there's a scene that perfectly illustrate your point in Sugar Coat Freaks, where Jill - the tsundere princess character, now with more blond twintails and less breasts - twists her ankle and blushes as the main character carries her.