I have to debate some contentions made by this question. It states that the US isn't a monarchy, and that Alice Roosevelt Longworth had no position in the administration. The question seems to imply a 'Royal' presence would have more impact on such a trip. It may have been true that she was occasionally a thorn in the side of her father (from the wiki page):
"I can either run the country or I can attend to Alice, but I cannot
possibly do both."
but she was still the daughter of the President of the United States, and on this trip was often treated as royalty.This article at SmithsonionMag seems to agree (emphasis mine):
She was, after all, the president’s daughter, which made her a
princess in all but title, and she conducted herself accordingly
The autobiography Crowded Hours has a couple of chapters dedicated to this trip, and this web page has many excerpts and photos. (All excerpts below from the web sites discussion of the book)
This particular passage relates Alice's reaction concerning being acknowledged alongside Japanese royalty:
“The Griscoms gave a garden party to which were invited all the
Americans in Tokio and Yokohama and all the Japanese Government
officials above a certain rank; also two pretty young Japanese
princesses, Nashimoto and Higashi-Fushimi—pretty even in occidental
dress. A matting was placed for them to stand on and I was escorted up
to stand beside them. All the Japanese women who approached them
curtsied, and then, to my astonishment, curtsied in my direction, too.
They only did it while I was standing on the matting with the
Princesses. The mere physical proximity to their venerated royalties
caused me to become, for the time it lasted, an object of respect. It
was a real "magic."”
Later in Canton where Americans weren't very welcome at the time:
Later we heard of lampoons that had been circulated in Canton, in
which I was pictured seated in a chair carried by four turtles. The
meaning of the writing around the picture was said to be very rude
indeed. It seems that in China to call one a turtle or to associate
one's name with a turtle is the equivalent of making a reflection upon
one's ancestry. We were told that the authors of the pamphlet would be
executed. So I had to intercede for them, to ask for mercy which I
believe was granted."”
I doubt executing the pamphlet authors would have been the response. If she wasn't considered either a direct representative Roosevelt and or the US, and requesting mercy on their behalf shows composure beyond that of some spoiled child on summer vacation.
Later, her observation concerning the Chinese Dowager Empress shows a definite political awareness of her trip.
“The Empress Tsz'e Hsi ranks with Catherine of Russia and Elizabeth of
England, with the Egyptian Queens Hatshepsut and Cleopatra, as one of
the great women rulers in history.
A later incident is related concerning a conversation with the Empress:
“The interpreter was Wu Ting Fang who had been Minister in
Washington. He stood between us, a little to the side, but suddenly,
as the conversation was going on, the Empress said something in a
small savage voice, whereat he turned quite gray, and got down on all
fours, his forehead touching the ground.”
“The Empress would speak; he would lift his head and say it in English
to me; back would go his forehead to the ground while I spoke; up
would come his head again while he said it in Chinese to the Empress;
then back to the ground would go his forehead again. There was no clue
to her reason for humiliating him before us.”
and later, showing she was not just 'along for the ride', she discussed this with her father, showing that she was indeed involved, if unofficially, in the politics of the trip:
When I told Father about it he thought it might have been to show us
that this man whom we accepted as an equal was to her no more than
something to put her foot on—
So even though parts of her book speak of Kipling and Marco Polo, showing the excitement she was feeling on this trip, the actions reported show she was treated with the respect and dignity afforded any foreign dignitary, and as daughter of the President was often treated as an American princess by those regions which could really only see her (or at least treat her) in the context of their own cultures. All this shows she was treated as an extension of the Roosevelt administration, whether or not she actually actually held any position.
Sorry for the book-length response.