Am I justifed in thinking that θηγέμεναι (= prose ἀκονᾶν) implies the spear was tipped with a metal tip rather than just being plain wood?
I suspect that ἀποξύνειν (or ἐξαποξύνειν) would be used for bringing a piece of wood to a point.
In many places in Homer it is clear that the spears (δόρυ) indeed had bronze (χαλκός) tips.
Il 5:72-75: τὸν μὲν Φυλεΐδης δουρὶ κλυτὸς ἐγγύθεν ἐλθὼν
βεβλήκει κεφαλῆς κατὰ ἰνίον ὀξέϊ δουρί:
ἀντικρὺ δ᾽ ἀν᾽ ὀδόντας ὑπὸ γλῶσσαν τάμε χαλκός:
ἤριπε δ᾽ ἐν κονίῃ, ψυχρὸν δ᾽ ἕλε χαλκὸν ὀδοῦσιν.
Indeed, Gaza has καλῶς μέν τις τὸ ἑαυτοῦ δόρυ ἀκονησάτω.