Which Messiah of Torah & Nevi'im did the Sanhedrin of 1-70 CE expect : The Anointed High Priest [Exodus 30, Leviticus 4] or The Son of David [1 Samuel 16:1-13, 2 Samuel 7:12-16, Jeremiah 22:5-6]?
The answer depends on the truth of [Exodus 30:30-33] and [2 Maccabees 2:1-8].
Which future "Messiah" required the anointing oil that Jeremiah hid from the Temple?
The appointed High Priest (required anointing oil hidden by Jeremiah) - This was not the expected Mashiach, since a High Priest is appointed individually & Kohanim cannot inherit the title. [Exodus 30 : v.30-33] - Although Based on Torah alone, the only “Messiah” | Ha-Mashiach הַמָּשִׁ֛יחַ ever mentioned by Moses in Leviticus 4 was a potential High Priest (Kohen Gadol) who atones for unintentional sins. - This title required the anointing oil.
Was David anointed with the same anointing oil as the Kohanim? Yes. - [1 Samuel 16:1-13]
Did David's prophesied son also require the anointing oil to inherit the everlasting Kingdom? - No. [2 Samuel 7:12-16]
Did The Son of King David as Messiah require priestly anointing oil (hidden by Jeremiah)? No. - [Jeremiah 22:5-6]
Pharisees & Sadducees were expecting "The Messiah" to be a "son of David" (a Messiah not requiring priestly anointing oil) in fulfillment of Jeremiah's prophecies (2 Samuel 7:12-16, JeremiahJeremiah 22:5-6, 2 Maccabees 2:1-8), not a "Son of God".