Our sages tell us, "there is no free man except one that involves himself in Torah learning" (see Avos 6:2).
If that's the case, why is Pesach called זמן חירותנו, a time of our freedom, when we did not have the Torah yet?
perhaps it is referred as that being the beginning of matan torah, and is called so for its purpose. as rav shimshon Raphael Hirsch says in "chorev" as to the meaning of sefiras ha'omer, that were counting to matan torah as that is the purpose of yetziras mitrayim.
Itamar Shwartz writes in his pesach talks book at bilvavi.net that our soul wants to do the right thing, but our yetzer hara and the fact that we're in exile hold us back. There's a force of evil that's external like an angel that persuades us to sin, and there's an evil that's within our hearts that causes our evil desires. The yetzer hara is mixed into our hearts. We view the outer force of evil as not being a part of us so we feel that we can overcome it, but we see the inner force of evil as a part of us, so it's much more difficult to overcome. Therefore, there's a yetzer hara within us, and a yetzer hara outside of us which causes us to sin. Additionally, there is a third kind of evil, the evil that the erev rav brought with them when we left mitzrayim. They brought an evil government with them. When Hashem spoke to us at har sinai, the erev rav didn't want to hear Hashem's voice, they wanted to hear it through moshe, and they were not ready to be moser nefesh for Hashem so they didn't hear His voice. When we heard the torah at har sinai, we attained freedom and it completed the freedom that began in mitzrayim. We had a certain freedom when we left mitzrayim, but also a certain freedom when we were given the torah. But still, the erev rav are there to prevent us from receiving holiness. When we left mitzrayim, our outer yetzer hara left us, and when we heard the torah the inner yetzer hara left us, but the erev rav wa still present. The erev rav didn't want us to be moser nefesh to hear the torah, and each year they come back to prevent us from being moser nefesh. In order to be free on pesach and be prepared for receiving the torah, we need to have mesirus nefesh. When it says there's no free person except the one who learns torah, it means that if we want to be free of the erev rav's influence, we need to have messirus efesh in learning torah. If we always yearn to do Hashem's will, it will help us have mesirus nefesh and have true cheirus to overcome the influence of the erev rav that persuades us.
See http://bilvavi.net/files/Pesach.Talks.pdf page 62-64
As noted here, when the rabbis say that a word means something in particular, in the form "X" is naught but "Y", they do not mean that this is the literal meaning of the word, but at most, that that is its usage in a particular context.
Moreover, as cited from R. Saul Lieberman z"l, it isn't necessarily even implying a strict interpretation in that context, but rather, it is a literary phrase used to introduce a Midrashic hyperbole.
Accordingly, the Mishna doesn't actually mean to actually interpret the word as חרות; freedom (in fact I don't think חרות even means freedom in Biblical Hebrew). And certainly doesn't mean that the only person who can ever be referred to as free in any context, is one who is occupied in Torah study.
In a similar vein, note the numerous references to freedom in Parashat Mishpatim (Exodus 21:2,5,26), none of which relate to Torah study.
That is the real answer.
Homiletically, we could answer that the time of the exodus is the time of freedom, since the point of the exodus was receiving thethe Torah. Cf. Exodus (3:12):
זֶה־לְּךָ֣ הָא֔וֹת כִּ֥י אָנֹכִ֖י שְׁלַחְתִּ֑יךָ בְּהוֹצִֽיאֲךָ֤ אֶת־הָעָם֙ מִמִּצְרַ֔יִם תַּֽעַבְדוּן֙ אֶת־הָ֣אֱלֹהִ֔ים עַ֖ל הָהָ֥ר הַזֶּֽה
This is the reason why I have sent you: when you remove the nation from Egypt, you will serve God on this mountain. (Translation following Ibn Ezra there)
The service at the mountain refers to receiving the Torah, as explained by R. Hayyim Paltiel, in his commentary there:
תעבדון את האלהים שתקבלו התורה
Since the point of the exodus was receiving the Torah, we an rightfully refer to the time of the exodus as the time of freedom, even were the term 'freedom' only appropriately applied to one occupied in Torah study, since the exodus was just the prelude to accepting the Torah, which is obviously the most essential prerequisite to being truly free by studying the Torah.