I believe that people with Advance Parole who are not paroled are subject to expedited removal. Like you said, they are inadmissible for trying to enter as immigrants (anyone not in nonimmigrant status is defined as an immigrant) without an immigrant visa. The definition of "arriving alien" in 8 CFR 1.2 and 8 CFR 1001.1(q), for expedited removal purposes, excludes people paroled before 1997, and people paroled after 1997 pursuant to an AP granted before they left, but neither of these exclusions cover people with AP who were not paroled, so the person in your case should still be an "arriving alien":
An arriving alien remains an arriving alien even if paroled pursuant
to section 212(d)(5) of the Act, and even after any such parole is
terminated or revoked. However, an arriving alien who was paroled into
the United States before April 1, 1997, or who was paroled into the
United States on or after April 1, 1997, pursuant to a grant of
advance parole which the alien applied for and obtained in the United
States prior to the alien's departure from and return to the United
States, will not be treated, solely by reason of that grant of parole,
as an arriving alien under section 235(b)(1)(A)(i) of the Act.
Interestingly, this definition of "arriving alien" was amended in 2006. Prior to 2006, the definition, last amended in 1998, was worded slightly differently:
An arriving alien remains such even if paroled pursuant to section
212(d)(5) of the Act, except that an alien who was paroled before
April 1, 1997, or an alien who was granted advance parole which the
alien applied for and obtained in the United States prior to the
alien’s departure from and return to the United States, shall not be
considered an arriving alien for purposes of section 235(b)(1)(A)(i)
of the Act.
On first reading, this seemed to say that someone who had an AP, that was granted before they left, would not be considered an "arriving alien" for expedited removal purposes (and thus not be subject to expedited removal), without regards to whether they were paroled. This would mean that the 2006 change substantively changed the situation for people with AP who were not paroled. However, the 2006 change said that the change to this sentence was just a "technical correction", to clarify the intended meaning of the 1998 change. On re-reading the 1998 sentence, I could maybe see a way it can be interpreted like the 2006 sentence, if the latter part of the sentence was all in the context of "even if paroled ... except", so it may all be referring to the case when the person was paroled; though I am not convinced. In any case, the 2006 text is in effect now, so it's hard to argue that they are not "arriving aliens".