I am less certain that "passage in question is not genuine", some of the objections are pretty good but the evidence on a whole is less weighty than the evidence against the Testimonium Flavianum. The late MS evidence must be checked by the attestation of the passage by Sulpicius Severus in the fifth century (Chronicle 2.29), who otherwise plagiarized much of Tacitus. The strongest evidence imho against its authenticity is the apparent silence by early church fathers, but this argumentum ex silencio is weakened by the fact that Tertullian does show a number of points of contact with Tacitus, Annals 15.44:
(1) In Apologeticus 5.3, Tertullian wrote: "Consult your own histories (commentarios vestros); you will there find that Nero was the first who wielded the sword of the Caesars furiously upon this sect [i.e. the Christianorum of 5.2], then first springing up at Rome (Romae)". Note that he here is telling his pagan readers to consult their own (= vestros) histories for a written statement about Nero persecuting the Christians in Rome. The only known statement in pagan sources that could account for this allusion is Annals 15.44. (2) In the same book (Apologeticus 16.1-3), Tertullian cited Tacitus in a hostile manner as the author of slanders against the Jews and Christians by extension (referring to Historia 5.2-6), so Tertullian confirms that Tacitus specifically lay before his mind as he wrote his apology. (3) In Annals 15.44, Tacitus states that the Christians were "hated for their vices (quos per flagita invisos)". Tertullian begins his apology by describing the hatred against Christians (Apologeticus 1.2-9), and then directly address the charge of illicit vices in 7.1-14 (cf. Justin Martyr, Apologia 1.26), one of which is that Christians like to "overturn lights and candles" (8.3; Justin Martyr mentions the same charge), which excellently links back to Tacitus, Annals 15.44 which claims that Nero "fastened the guilt" of causing the fire to the Christians. (4) Annals 15.44 also traces the origin of the Christianos to "the reign of Tiberius" (Tiberio imperitante) when Christus was executed, and Tertullian similarly said that "very many know as we declare that it [our sect] is relatively new, originating in the time of Tiberius (Tiberiani temporis)" (Apologeticus, 21.1). So it seems quite plausible to me that Tertullian knew of the passage in the Annals, even if he does not quote it directly.
As for other church fathers not citing it as a witness to Christ, the offensive slurs contained within it (e.g. that the faith is a "pernicious superstition," that it is an "evil", that it is "horrible and shameful", that Christians practice illicit vices) may well explain the silence; Tertullian seems to have used it mainly to respond to the statements.
What makes me lean more towards judging the passage as authentic is the (1) similarity in literary style with the rest of Tacitus, (2) the lack of seams or breaks in the narrative by the passage (which is the natural climax of the narrative concerning propitiation), (3) the unlikelihood that a Christian interpolator would put in Tacitus' mouth various slurs about the Christians and also fail to make Tacitus report the resurrection, and (4) the similar views expressed elsewhere in Tacitus.
For an instance of the latter, Annals 15.44 states that "the origin of the evil (originem eius mali)" is "in Judea" and the capital Jerusalem is "where all things horrible and shameful (atrocia aut pudenda) collect and are practiced (celebranturque)". In Historia 5.2-3, Tacitus mentions that the Jews were "a race detested by the gods" and had their origins in Egypt as a race of disease-ridden people expelled by the Egyptian king "to cleanse his kingdom" (cf. Manetho). Moses then led these filthy people to Judea where they "founded a city and a temple". And then Tacitus wrote that Moses "gave them a novel form of worship, opposed to all that is practiced by other men. Things sacred with us have no sanctity with them, while they allow what is forbidden by us" (5.3), such that "all their other customs are at once perverse and disgusting (sinistra foeda) and owe their strength to their very crookedness (pravitate)...The Jewish religion is sordid and absurd (absurdus sordidusque)" (5.4). The attempt to describe a shameful origin for the Jews is also reminiscent of the similar attempt in Annals 15.44 with respect to the Christians.
As for the historicity of the description of the Neronic persecution, since the passage otherwise shows that the author relied on Christian sources (e.g. the reference to "Christ" rather than "Jesus"), the probable exaggeration of referring to the matryrs as an "immense multitude" may reflect later Christian storytelling (cf. Revelation 7:14 which refers to an "innumerable multitude" of martyrs, 1 Clement 6:1 which refers to the "vast multitude of the elect" who died for their faith), or it may reflect literary license. On the other hand, if Nero had Christians rounded up in other provinces, then a rather large number could well have been executed. A Neronic persecution provides excellent background for the allusions to past martyrs and Nero in Revelation, and 1 Clement 6:1-3, written probably in Rome, seems to also reflect a past bloody persecution (e.g. the reference to Danaids and Dircae, which links to the mythological plays mentioned in Tacitus, Annals 15.44.7 and Suetonius, Nero 11.2, 12.2). See also Ascension of Isaiah 4:2-3, drawing on a tradition about Nero persecuting the church. The slur that Christians overturn lamps and maliciously set fires (reported by Justin Martyr and Tertullian) also is neatly explained as a reflexion of Nero's charge as reported by Tacitus.
I don't regard the matter entirely clear either way, but I believe that there is good reason for treating Annals 15.44 as written by Tacitus.