"Does anyone actually learn Torah directly following Birchas HaTorah at
the beginning of Shachris?"
Yes! For general halachic context to studying before Shacharit see:
For a similar discussion here on Mi Yodeya, see: Learning before Davvening
"Common sense would dictate that right after we thank HaShem for
giving us the great Mitzvah to learn Torah we should learn it."
Some agree with your premise, and others don't see a need to actually open a book and study. See Rav Eliezer Melamed's answer in Peninei Halacha
http://ph.yhb.org.il/en/03-10-05/ (partially quoted below):
"Birkot HaTorah recited in the morning cover all of one’s learning throughout that day, and any work or business conducted in the interim is not considered to be an interruption (Shulchan Aruch 47:10).
The Rishonim disagree concerning the question of whether it is an obligation to learn a few verses of Torah immediately upon reciting Birkot HaTorah. According to Ri, one of the Ba’alei HaTosafot (Berachot 11b, “Shekvar”), Birkot HaTorah differ from the other Birkot HaMitzvot in that they are not directed only towards the present learning, rather they are designated for Torah learning throughout the entire day. Therefore, there is no obligation to learn specifically after the berachah, rather the obligation is to learn something during the day. That is how the Beit Yosef interprets the opinions of the Rosh and the Tur as well. However, the Rambam maintains that the law concerning Birkot HaTorah is like all Birkot HaMitzvot, in which it is necessary to adjoin the berachah to the mitzvah and therefore one must learn immediately following its recital. If he does not learn immediately, the berachah becomes nullified. This case is similar to a person who wants to eat cake and recites, “Borei minei mezonot”, yet does not eat right away, and instead goes to do other things. When he finally wants to eat from the cake, he will need to go back and repeat the Mezonot blessing. Even though it seems from the Shulchan Aruch that the ruling is according to Ri, nonetheless, in the opinion of many Acharonim we follow the Rambam (Mishnah Berurah 47:19), for the Rama in Darkei Moshe explains the opinions of the Rosh and the Tur like the Rambam. Nowadays, the prevalent minhag among all Jews is to recite the three verses of Birkat Kohanim after Birkot HaTorah and that is considered learning after the berachah. After that, many add other rabbinic words such as “Eilu devarim…”, for they contain words of Mishnah and Beraita, (which are considered words of Gemara), and in that way every Jew merits learning Scripture, Mishnah, and Gemara daily."