Who owns copyright of a photo taken by a camera after the timer is set to 10 seconds? Is it the person who set the timer? But he/she didn't take any picture. A machine did.
To the extent that the photograph is an expression of the operator's originality (e.g. framing, subject, angles, camera settings, etc.) it is the operator who has the best claim to copyright. Whether it was "made by a machine" misses the issue: what matters is the degree to which the purported author contributed originality.
In your hypothetical, the operator is the person responsible for fixation of the work and the person who contributed original expression to the work. It is the "the product of [that person's] exercise of skill and judgment" (the Canadian standard) and an expression of a "modicum of creativity" (the U.S. standard). See generally, CCH Canadian Ltd. v. Law Society of Upper Canada, 2004 SCC 13; Feist Publications, Inc. v. Rural Telephone Service Co. Inc., 499 U.S. 340 (1990).
But the Copyright Office has stated that:
[it] will not register works produced by a machine or mere mechanical process that operates randomly or automatically without any creative input or intervention from a human author. The crucial question is “whether the ‘work’ is basically one of human authorship, with the computer [or other device] merely being an assisting instrument, or whether the traditional elements of authorship in the work (literary, artistic, or musical expression or elements of selection, arrangement, etc.) were actually conceived and executed not by man but by a machine.”
Copyright Compendium, s. 313.2
I am comfortable predicting that a mere 10-second delay will almost always fall on the side of the line where the operator would be considered to be an author.1
1. At the other end of the spectrum are pre-positioned cameras that record continuously or when triggered by motion. Who (if anyone) holds the copyright in that circumstance has not been answered, with opinions differing across jurisdictions and academics: some arguing that the device owner would hold copyright, some arguing that no one would, others arguing that it would be fact-based inquiry about who might have contributed sufficient originality.
The camera itself is a machine, the timer is a functional mode of the machine. The camera also has other functional modes that often factor into the process of taking a photograph, including autofocus, automatic exposure control, flash, tripods for stabilization, etc.
If the photographer chose to NOT use a timer, the photograph would still have been taken by a machine. However, the decision to use a timer or not is as much a part of the creative process as using any other mode.
Copyright laws do not go into detailed requirements of specific camera settings needing to be controlled manually or directly by the photographer in order to have ownership of copyrights. Therefore, if a timer was used the copyright would belong to the photographer just as it would if a timer was not used.
If there were a controversy over participation by a third party in any aspect of the process it would have to be settled in court.
Assuming the photo is eligible for Authors' Rights at all, the photographer owns the Author's Rights.
An important part of Author's Rights is creative choice. The photographer chose to set the timer to 10 seconds. They could have chosen 5 seconds or 15 or not to use a timer at all. Same as the subject, composition, framing, depth-of-field, focus, shutter speed, focal length, lens, body, whether to use digital (and which sensor size) or film (and which film), etc.
The camera didn't make those choices. The photographer did.
By the OP's argument, any author who uses a computer to write their novels does not own the copyright, because the novel was written by the computer. Any painter who uses a brush does not own the copyright, because the painting was painted by the brush. Any composer who uses a guitar does not own the copyright, because the song was played by the guitar.