There were seats drawn on the blue print for the tower. Today there are no seats. Were they fitted and then taken out or were they never installed at all?
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It is unfortunately far easier to prove the presence of seats than their absence. The newspapers worldwide covered the Eiffel Tower in great detail and I decided to check what they wrote about the elevators (which where obviously a particular point of interest). The National Library of New Zealand puts historical newspapers online and makes them searchable, that's what I looked at. One typical description comes from the Otago Daily Times, 14 December 1889:
The elevator is a handsomely furnished room about 10 x 10 [feet?], raised by steam power, without any perceptible jarring or motion. The visitors can be stopped at either of the three stages, but to enjoy this serial journey the last is doubtless the altitude all should go to. From this roomy platform, upon which seats have been provided, a most enchanting panorama is seen.
Note how it mentions seats on the platform but not in the elevator. From Te Aroha News, 28 September 1889:
There is but one lift from the second to the third platform, and this works in two stages, and 170 people can be taken up at one time.
Numbers in these articles have to be taken with a grain of salt of course, I've seen articles published before the tower was opened talking about estimated numbers of 5000 people transported per hour which is clearly very far off. But seating 170 people would make an awfully large lift. And finally, Otago Witness, 24 October 1889:
Every scrap of iron or wood within reach upon the Eiffel Tower is completely covered with names and dates. The interiors of the lifts are covered, and the glass wind-protectors of the elevators on the various floors are rapidly being filled up.
Again, seats would be a good candidate to be mentioned, not only the wind-protectors. But this article doesn't talk about them.
Having decided from this indirect evidence that seats on the original elevators were unlikely I found the 2010 book Elevator Systems of the Eiffel Tower, 1889 (that's apparently the book where your illustration comes from). And judging from this book, your graphic refers to the elevator with the smallest capacity (in terms of persons transported), by Otis:
The double car, traveling at 400 feet per minute, carried 40 persons, all seated because of the change of inclination.
Oops! About the French elevators by Roux the book says:
Capacity was equivalent to that of the Otis system. About 100 people could be carried in the double-deck cabin, some standing.
And the two stage elevator mentioned in Te Aroha News was built by Edoux and had a capacity of 60 persons. The book doesn't tell whether people were seated in this elevator and the illustration sadly doesn't help because it shows the passengers while they are changing elevators midways.
However, the official guide of the Eiffel Tower (printed in 1893) says the following about this elevator:
Les cabines, fort élégantes, pourvues de sièges articulés et de banquettes, ont 14 mètres carrés
The cabins, very elegant, fitted with hinged seat and benches have 14 square meters
And in fact, in this photo of this elevator you can clearly see that the women on the right are sitting. So to answer your question: yes, originally most people indeed took a seat in the Eiffel Tower elevators.