Thanks for the great question. I have watched much Rabbi Friedman and have spent countless hours collecting sources on the topics he lectures on.
I don't really need to bring those, because the general way you've stated the question isn't really something that he specifically says (although he does focus on it), but a general principle of Torah.
The purpose of free will is so that we can keep Torah, and your source demonstrates that well. As the Rambam puts it in Hilchot Teshuva 5:
Every person has the ability to lead a good life and be a tzaddik or
to lead an evil life and be a rasha... This is a fundamental concept
underlying the whole Torah and its mitzvot. Moshe said (Nitzavim,
D'varim 30:15), "Look, I have put before you today a choice of life
and good or death and evil." He said as well (Re'eh, D'varim
11:26-28), "See, I put before you today blessings and curses;
blessings if you obey... and curses if you do not obey God's
Mitzvot..." In other words, you have the ability to choose between all
types of human action, whether good or bad... If this were not so
what place would there be for the Torah; with what justice could God
punish the rasha or reward the tzaddik if each does not have perfectly
free will to choose his own path?
The 613 mitzvot are the Divine service, as it is written in Dvarim 10:12-13:
וְלַֽעֲבֹד֙ אֶת־ה' אלוקיך בְּכׇל־לְבָבְךָ֖ וּבְכׇל־נַפְשֶֽׁךָ׃ לִשְׁמֹ֞ר אֶת־מִצְוֺ֤ת ה' וְאֶת־חֻקֹּתָ֔יו אֲשֶׁ֛ר אָנֹכִ֥י מְצַוְּךָ֖ הַיּ֑וֹם
to serve your God with all your heart and soul, keeping Hashem’s
commandments and laws, which I enjoin upon you today
See also Ibn Ezra on the passage we say in Shema about serving Hashem. See also this Rabbeinu Bayha on how we were created only to serve Hashem.
The Torah is the "instruction" (that's what the word torah means) in serving God. It is essentially synonymous with Mitzva (Shemot 24:12):
וַיֹּ֨אמֶר ה' אֶל־מֹשֶׁ֗ה עֲלֵ֥ה אֵלַ֛י הָהָ֖רָה וֶהְיֵה־שָׁ֑ם
וְאֶתְּנָ֨ה לְךָ֜ אֶת־לֻחֹ֣ת הָאֶ֗בֶן וְהַתּוֹרָה֙ וְהַמִּצְוָ֔ה
אֲשֶׁ֥ר כָּתַ֖בְתִּי לְהוֹרֹתָֽם׃
Hashem said to Moses, “Come up to Me on the mountain and wait there,
and I will give you the stone tablets with the Torah and mitzvot
which I have inscribed to instruct them [same root as תורה].”
Torah study itself is also a mitzva, a Divine service. Prayer, another mitzva according to some, is the specific meaning of Avodah, service, whereas the rest of the Mitzvot are the more general meaning. As Rambam puts it in Sefer HaMitzvot:
הוא שצונו לעבדו, וכבר נכפל זה הצווי פעמים באמרו ועבדתם את ה' אלקיכם,
ואמר ואותו תעבודו. ואע"פ שזה הצווי הוא גם כן מהצוויים הכוללים כמו
שביארנו בשורש ד', הנה יש בו יחוד אחר שהוא צווי לתפלה.
The 5th mitzvah is that we are commanded to serve G‑d (blessed be He).
This commandment is repeated twice. It is stated: "And you shall serve
G‑d, your L‑rd" (Exodus 23:25). And it is also stated: "And you shall
serve Him" (Deuteronomy 13:5). Although this commandment is of a
general nature, as explained in the Fourth Principle, [and apparently
should not be included in the count of the 613 mitzvos,] nevertheless
it has a specific quality, since it is the commandment to pray.
Prayer is service of the heart.
All the above now connects back to free will in the pasuk Shmuel quoted in his comment:
See, I have set before thee this day life and good, and death and evil: in that I command thee this day to love the Lord thy God, to walk in His ways, and to keep His commandments and His statutes and His judgments (Devarim 30:15-17)
and ibid 19:
Regarding Moshiach, Rabbi Friedman's main position is that of the gemara in Sanhedrin 98a, that the Mosiach will come
בדור שכולו זכאי או כולו חייב
in a generation that is entirely innocent or entirely guilty
He happens to be very optimistic about this speculative point, as was the Lubavitcher Rebbe T'Zl (and I remember a beautiful lecture by Rabbi Lawrence Kelemen on this too), that our generation is indeed כולו זכאי and there are beautiful explanations of why. I have heard him mention that maybe it means most of the world - he is an optimist, as I said!
As for the purpose of the world, yes, that's one way to put it. Sandhendrin 98b:
אמר רב לא אברי עלמא אלא לדוד ושמואל אמר למשה ורבי יוחנן אמר למשיח
Rav says: The world was created only for David. And Shmuel says for
Moshe. And Rabbi Yoḥanan says: for Moshiach.
The idea Rabbi Friedman discusses a lot regarding this is based on a couple of midrashim that say the following (e.g. Midrash Tanchuma Nasso 16:1)
נִתְאַוָּה שֶׁיְּהֵא לוֹ דִּירָה בַּתַּחְתּוֹנִים
Hashem desired a dwelling place in the lowest [world]
This is first brought in Tanya and developed heavily in Chabad writings and is very connected to the general idea you've brought here. It is the beginning of a beautiful focus in thought on what the world and we mean to Hashem, i.e. a Torah on looking at things from His point of view. If you wish to learn more about it, Rabbi Friedman has literally thousands of videos on it, and is great at explaining them.