Ven Thanissaro mentions at least one of them in the essay you qoute,
Whatever ascetics and brahmins theorize about the future, and propose various hypotheses concerning the future, all of them propose one or other of these five theses.
Ye hi keci, bhikkhave, samaṇā vā brāhmaṇā vā aparantakappikā aparantānudiṭṭhino aparantaṁ ārabbha anekavihitāni adhivuttipadāni abhivadanti, sabbe te imāneva pañcāyatanāni abhivadanti etesaṁ vā aññataraṁ.
this quote very oddly occurs after only 4 of the theories have actually been discussed, 3 kinds of identity after death views and 1 kind of annihilationist view, skipping over the 5th listed in the start of the sutta, that is extinguishment views.
Just this is enough to show that the text is corrupt.
later in the discussion of progressively letting go of grasping we have;
Now, some ascetics and brahmins, letting go of theories about the past and the future, shedding the fetters of sensuality, go beyond the rapture of seclusion, spiritual bliss, and neutral feeling.
Idha pana, bhikkhave, ekacco samaṇo vā brāhmaṇo vā pubbantānudiṭṭhīnañca paṭinissaggā, aparantānudiṭṭhīnañca paṭinissaggā, sabbaso kāmasaṁyojanānaṁ anadhiṭṭhānā, pavivekāya pītiyā samatikkamā, nirāmisassa sukhassa samatikkamā, adukkhamasukhāya vedanāya samatikkamā:
They regard themselves like this: ‘I am at peace; I am extinguished; I am free of grasping.’
‘santohamasmi, nibbutohamasmi, anupādānohamasmī’ti samanupassati.
The Realized One understands this as follows.
Tayidaṁ, bhikkhave, tathāgato abhijānāti.
This good ascetic or brahmin, letting go of theories about the past and the future, shedding the fetters of sensuality, goes beyond the rapture of seclusion, spiritual bliss, and neutral feeling.
Ayaṁ kho bhavaṁ samaṇo vā brāhmaṇo vā pubbantānudiṭṭhīnañca paṭinissaggā, aparantānudiṭṭhīnañca paṭinissaggā, sabbaso kāmasaṁyojanānaṁ anadhiṭṭhānā, pavivekāya pītiyā samatikkamā, nirāmisassa sukhassa samatikkamā, adukkhamasukhāya vedanāya samatikkamā:
They regard themselves like this: ‘I am at peace; I am extinguished; I am free of grasping.’
‘santohamasmi, nibbutohamasmi, anupādānohamasmī’ti samanupassati;
Clearly this venerable speaks of a practice that’s conducive to extinguishment.
addhā ayamāyasmā nibbānasappāyaṁyeva paṭipadaṁ abhivadati.
Nevertheless, they still grasp at theories about the past or the future, or the fetters of sensuality, or the rapture of seclusion, or spiritual bliss, or neutral feeling.
Atha ca panāyaṁ bhavaṁ samaṇo vā brāhmaṇo vā pubbantānudiṭṭhiṁ vā upādiyamāno upādiyati, aparantānudiṭṭhiṁ vā upādiyamāno upādiyati, kāmasaṁyojanaṁ vā upādiyamāno upādiyati, pavivekaṁ vā pītiṁ upādiyamāno upādiyati, nirāmisaṁ vā sukhaṁ upādiyamāno upādiyati, adukkhamasukhaṁ vā vedanaṁ upādiyamāno upādiyati.
And when they regard themselves like this:
Yañca kho ayamāyasmā:
‘I am at peace; I am extinguished; I am free of grasping,’ that’s also said to be grasping on their part.
‘santohamasmi, nibbutohamasmi, anupādānohamasmī’ti samanupassati tadapi imassa bhoto samaṇassa brāhmaṇassa upādānamakkhāyati.
So we have brahmins who have “let go of theories about the past and the future, shedding the fetters of sensuality, go beyond the rapture of seclusion, spiritual bliss, and neutral feeling.”
but who " Nevertheless, they still grasp at theories about the past or the future, or the fetters of sensuality, or the rapture of seclusion, or spiritual bliss, or neutral feeling."
This cannot be right, either those things are let go or not, what seems to be implied here is that while such talk is the “right talk” we nevertheless infer that the Brahmins have not in fact let go of such grasping because of their “I am this” and “I am that” chest beating and pompousness.
The sutta is also a bit odd in it’s structure in that the Buddha is speaking the main body of the text but then the the Buddha quotes what they themselves are saying and says;
The Realized One understands this as follows.
Tayidaṁ, bhikkhave, tathāgato abhijānāti.
So the text is a sort of suttavibhanga type of analysis, where the parts of a well known text are expanded and explained, but here there is not a great deal of expansion, merely the phrase
‘All that is conditioned and coarse. But there is the cessation of conditions—that is real.’
‘Tayidaṁ saṅkhataṁ oḷārikaṁ atthi kho pana saṅkhārānaṁ nirodho atthetan’ti—
Understanding this and seeing the escape from it, the Realized One has gone beyond all that.
iti viditvā tassa nissaraṇadassāvī tathāgato tadupātivatto.
applied to each position, also the Buddha does not say , “that is what I said, but why did i say it” which is more usual for suttavibhanga suttas where the Buddha is the speaker.
the sphere of nothingness is also “squeezed in” to an analysis part but isn’t mentioned in the original exposition part.
There are probably a lot of other things that should cause alarm, but my Pali is too poor to see the not so obvious ones.
Fundamentally I think that when read in conjunction with MN1 and DN1 and DN2 and DN9 this sutta is part of the most fundamental philosophical bedrock of the Buddhas teaching, touching on the most subtle and profound aspects of anatta or conditionality by exploring how other philosophical positions give rise to difficulties and imperfections, but it is jumbled up, and difficult to decipher, so like @sujato says, it should be approached with caustion.
Metta