The ruling regarding Salat-ut-Tasbeeh is that it is permissible and, in fact, very meritorious.
The Hadith in question has been accepted as Sahih (authentic) or Hasan (sound) by various eminent Masters of Hadith. Those that claimed that it was weak did so on the basis of a few chains of the narration. However, if all the various chains be gathered, there will remain no doubt as to its authenticity.
Hafiz ibn Hajar al-Asqalaani (RA) states that sometimes a Muhaddith classifies a certain Hadith as weak, very weak or even as a fabrication based on one or two chains that were available to him, whereas there may be other chains through which that Hadith may be classified as Hasan (sound) or even Sahih (authentic). (Anukat vol.2 pg.848-850)
Hafiz ibn Hajar (RA) then mentions the Hadith of Salat-ut-Tasbeeh as an example for this and he accepts it to be in fact either Sahih (authentic) or Hasan (sound) and not Dha'eef (weak). (Ibid)
Imaam Tirmidhi (RA) says that many Ulama, among them Imaam Abdullah ibn al-Mubaarak (RA), have accepted the virtue of Salat-ut-Tasbeeh. (Tirmidhi vol.2 pg.348; Hadith 481)
Imaam Bayhaqi (RA) states that ' Salat-ut-Tasbeeh was the practice of Abdullah ibn al-Mubaarak and may pious predecessors of various eras. And this in fact lends strength to its acceptability.' (Shu'ubul Imaam vol.1 pg.427; Ilmiyyah)
Allaamah Munzhiri (RA) mentions: Many Muhadditheen have accepted it's authority from among them are: 1) Imaam Abu-bakr al-Ajurriy, 2) Imaam Abu Muhammad al-Misriy (Ustaadh of Allaamah Munzhiri), 3) Hafiz Abul-Hasan Maqdisi (Ustaadh of Allaamah Munzhiri), 4) Imaam Abu-Dawud and 5) Imaam Haakim. (Targheeb vol.1 pg.468)
Allaamah Suyuti (RA) has enumerated upto 20 great Muhadditheen who have accepted its authenticity. Besides those that are mentioned above, some of them are: 6) Hafiz Abu-Sa'eed al-Sam'aaniy, 7) Hafiz Khateeb al-Baghdaadiy, 8) Hafiz ibn-Mandah, 9) Imaam Bayhaqi, 10) Imaam al-Subkiy, 11) Imaam al-Nawawiy, 12) Haafiz ibn al-Salat, 13) Hafiz Abu-Musa al-Madiniy, 14) Hafiz al-Alaaeiy, 15) Imaam Siraaj-ud-Deen al-Bulqiniy, 16) Hafiz al-Zarkashiy and a few others. (al-Laalil Masnoo'ah vol.2 pg.42-44)
The following list of Muhadditheen is of those who have written detailed articles on this matter: 1) Imaam al-Daraaniy, 2) Imaam Abu-Musa al-Madiniy,
3) Imaam ibn Mandah, 4) Hafiz ibn Naasirud-Deen ad-Dimishqiy, Allaamah al-Suyyooti.
As for the claim of the writer, 'It had been unknown to the great Imaams. and presumably Imaam Shaafi'ee (RA).'
This is a claim that lacks the support of explicit quotations from those illustrious Imams as well as any reference. In fact, the books of Hanafi Fiqh support the view of its acceptance. (refer Shaami vol.2 pg.27; HM Saeed). And several Shaafi'ee Jurists have also endorsed it, namely Imaam al-Muhaamiliy, Imaam al-Juwaini, Imaam-ul-haramayn, Imaam Ghazaaliy, Imaam Raafi'ee and others. (al-La-aaliy vol.2 pg.43; al-Azkaar of Imaam Nawawiy pg.242)
Khateeb Baghdaadiy (RA), who is Maaliki, states that there is no reason for it not being permissible. In fact, Imaam ibn Hajar (RA) has mentioned a quotation from Imaam Maaliki from which it could be deduced that it was acceptable in his Madhab. (refer Futuhaat al-Rabbaaniyyah vol.4 pg.321) Qaadhi Iyaad Maaliki (RA) has also accepted its virtue. (Ibid)
As far as Imaam Ahmad ibn Hanbal (RA) is concerned, Hafiz ibn Hajar (RA) has stated that he had reversed his decision and was possibly inclined towards its acceptability later on. (Futuhaat al-Rabbaaniyyah vol.4 pg.318, 320)
From the above, it is apparent that there can be no doubt regarding Salat-ut-Tasbeeh being an act of virtue which has been established authentically. And is accepted by a large number of celebrated scholars of Hadith as well as the scholars of Fiqh (jurisprudence).
Imaams Taajud-Deen Subki (RA) and Badrud-Deen Zarkashi have stated that it is from the fundamentals of Deen and anyone who discards it despite knowing its virtue is not on the correct path. (Futuhaat vol.4 pg.321-322)
And Allah Ta'ala Knows Best