Tony Chan's answer made me realize that it would make sense to look at other usages of this verb in the NT. These seem to fall into various groups.
(1) There is the passage I asked about, Mark 2:25, which is also duplicated in Matthew 12:3 and Luke 6:3. These seem ambiguous to me.
(2) There are a bunch of places where the context makes it clear that someone is reading a written text, e.g., John 19:20 (reading the sign that says "JESUS OF NAZARETH, THE KING OF THE JEWS") and Acts 15:31 (reading a letter that has just been delivered). These make it certain that at least some usages of the verb in the NT do refer to reading a written text. (a) In John 19:20, nobody is reciting the text of the sign out loud, whereas (b) in most of the other examples someone is reading out loud to other people.
(3) And finally there are some cases where scripture is being recited out loud on the sabbath, such as Acts 13:27 (all translations from WEB):
For those who dwell in Jerusalem, and their rulers, because they didn’t know him, nor the voices of the prophets which are read every Sabbath, fulfilled them by condemning him.
And Acts 15:21:
For Moses from generations of old has in every city those who preach him, being read in the synagogues every Sabbath.
He came to Nazareth, where he had been brought up. He entered, as was his custom, into the synagogue on the Sabbath day, and stood up to read. The book of the prophet Isaiah was handed to him. He opened the book, and found the place where it was written, ...
The examples in Acts 13:27 and Luke 4:16 describe rabbis reciting out loud to an audience. The audience of Nazarean peasants in Luke 4:16 would have understood only Aramaic, not Hebrew, and likely the same would have been true for the audience envisioned in Acts 13:27. Targums were only oral during Jesus's lifetime -- it was forbidden to write them down. In Luke 4:16, there is a clear implication that the people in the room understood Jesus. So if Luke is accurately describing something that actually did take place, then it seems that ἀναγνῶναι would have to describe something that is not just the idea of someone reading a written text out loud, verbatim. Jesus may be paraphrasing in Aramaic on the fly, and referring to the text to remind himself of what it says. Or possibly he reads the Hebrew out loud, which almost nobody in the room can understand, and then follows up with an Aramaic translation.
So it seems clear to me that the word is used with a variety of meanings in the NT, and that among these meanings are (2a) to silently read a written text to one's self, (2b) to read a text out loud to others, and (3) to remind people of certain words (the literal meaning of ἀνα + γιγνώσκω), in a context where it's not possible that the reminder is a verbatim reading out loud from a piece of writing.