You say that the Greek King Milinda observed Brahman rites:
", In the land of the Bactrian Greeks, there was a city called Sagala, a great centre of trade. Rivers and hills beautified it, delightful landscapes surrounded it, and it possessed many parks, gardens, woods, lakes and lotus-ponds. Its king was Milinda, a man who was learned, experienced, intelligent and competent, and who at the proper times carefully observed all the appropriate Brahminic rites, with regard to things past, present and future. As a disputant he was hard to assail, hard to overcome, and he was recognized as a prominent sectarian teacher"
Well, I looked at the on-line book you linked to and here is what it says in the intro:
"The question then arises whether the personages were any more real than the conversations. Milinda is supposed to be the Menander, who appears in the list of the Greek kings of Baktria, since he is described in the book as being a king of the Yonakas reigning at Sāgala (the Euthydemia of the Greeks), and there is no other name in the list which comes so near to Milinda. "
So the connection to the Greeks is weak and even suspicious. Later on the same intro says this:
"Then in the title of the lost forty-first book of Justin's work, Menander and Apollodotus are mentioned as 'Indian kings.'"
However, there seems to be some 'hard' evidence that he was Greek:
"The only remaining ancient evidence about Menander-Milinda (apart from what is said by our author himself), is that of coins. And, as is usually the case, the evidence of the coins will be found to confirm, but to add very little to, what is otherwise known..... All the coins have a legend in Greek letters on one side, and a corresponding legend in Ariano-pāli letters on the other side. ....To sum up.--Menander-Milinda was one of those Greek kings who carried on in Baktria the Greek dominion founded by Alexander the Great. "
But the intro also claims the he was secular and suggests that the Pali author made up his Bramanism:
"When our author says that Milinda, was converted to Buddhism 3, he may be either relating an actual tradition, or he may be inventing for his own purposes.....none of Menander's coins show any decisive signs of his conversion"
Well, anyway what is important for our discussion is that Melinda, if he was a Greek King who stayed on in India after Alexander's death, was several generations after the Greek philosophers invented the concept of the soul. So when Alexander and other Greek Kings brought that concept into India, it was rejected by the Buddhists, probably for political reasons. The remaining question is how the word 'soul' got into sutras that were presumably written much earlier. Had that concept already spread to India from Greece during the time of Gautama.
Gautama died in 483BC. Alexander died in 333BC. So this tale was at least 150 years after Gautama's death. So when did the concept of the soul originate in Greece. Socrates talks of the soul, but he died in 393BC, 100 years later than Gautama. The first Greek philosopher Thales died in 547Bc, but he more mathematician than philosopher. But apparently the idea of soul originated with him. Here are some quotes from
'Aristotle de an. A5, 4II a7 (1)
And some say that it [soul] is intermingled in the universe, for which reason, perhaps, Thales also thought that all things all full of gods.'
'Diogenis Laertius I,24 (DK ii Ai) (1)
According to Thales the soul is immortal'
'Aetius I,7, II (1)
Thales said that the mind of the world is god, and that the sum of things is besouled, and full of daimons; right through the elemental moisture there penetrates a divine power that moves it.'
So it is entirely possible that the concept of the soul was known to Gautama, who was alive some 60 years after Thales. But somehow I suspect that it did not become an issue until Alexander invaded India. And so I suspect that the Sutras were modified for political reasons at that time.