It seems that σπεύδω has the meaning of "hasten" or "press on" when intransitive (i.e. is without object), but can mean "seek eagerly" or "strive after" something when transitive (i.e. with object). This is clarified in some lexicons (e.g. Liddell-Scott-Jones, Barclay), but not others (e.g. Swanson's Dictionary of Biblical Languages). The confusion lies in that when meaning "hasten" the Greek verb (as the English) can be either transitive or intransitive. One could oneself be hurrying, or one could be hurrying someone along.
Examples of σπεύδω in the intransitive:
Καὶ ἦλθον σπεύσαντες, καὶ ἀνεῦρον τήν τε Μαριὰμ καὶ τὸν Ἰωσήφ, καὶ τὸ βρέφος κείμενον ἐν τῇ φάτνῃ.
And they went with haste [lit. they went hurrying], and found Mary and Joseph, and the babe lying in a manger.
Σπεῦσον καὶ ἔξελθε ἐν τάχει ἐξ Ἱερουσαλήμ· διότι οὐ παραδέξονταί σου τὴν μαρτυρίαν περὶ ἐμοῦ.
Make haste and get quickly out of Jerusalem, because they will not accept your testimony about me.
2 Peter 3:12 seems to be the only place in the New Testament where σπεύδω is used in the transitive (there are 6 usages total), but it is used this way in places in the Septuagint, for example:
σπεῦσον καιρὸν καὶ μνήσθητι ὁρκισμοῦ, καὶ ἐκδιηγησάσθωσαν τὰ μεγαλεῖά σου.
Await the day, and remember the appointed time, and let people recount thy mighty deeds.
Susanna 12 (prelude to Daniel in LXX)
καὶ ὡς ἐγίνετο ὄρθρος, ἐρχόμενοι ἔκλεπτον ἀλλήλους σπεύδοντες, τίς
φανήσεται αὐτῇ πρότερος καὶ λαλήσει πρὸς αὐτήν.
And they watched eagerly, day after day, to see her.
There are also numerous examples given by Liddell-Scott-Jones in ancient Greek literature (e.g. Thucydides' Peloponnesian War 5.16, Euripides' Hecuba line 1175) where it is used in the transitive sense of "await eagerly".
The Orthodox New Testament (done by Greek Orthodox nuns) translates 2 Peter 3:12 as expecting and earnestly desiring the coming of the day of the Lord, but it seems, as you say, most modern translations seem to imply that one can somehow "hasten the day".