Your average Jews in Temple times did not go participate actively in the regularly scheduled Temple offerings. For them, probably the biggest sacrificial experience of the year was the Korban Pesach on 14 Nissan afternoon. It is one of the two positive commandments in the Torah (along with Brit Milah) that have a punishment of Karet (Mishna Kereitot 1:1), women are obligated in it despite its being a positive time bound commandment (Rambam KP 1:1), it has a second opportunity to be brought (Numbers 9), and seems in Tanakh to be a central part of a Jew's religious experience. 14 Nissan was seen as a holiday, with its own prohibition on Chametz (Pesachim 5a), a prohibition on doing labor (Pesachim 50), and Hallel being recited in the Temple (Pesachim 64) accompanied by flutists (Arachin 10).
The Talmud (Brachot 26b) tells us that the prayer services enacted by Chazal were based on the Temple sacrifices (hence, the existence of an extra "Musaf" prayer on days when there was a Korban Musaf).
Why did Chazal not enact a liturgical commemoration of the Korban Pesach on 14 Nissan? There could have been an extra Amida said after Mincha (a pseudo-Musaf). There could have been a Torah reading (with Haftarah?) at Mincha. There could have been a public Hallel.
(I'm aware that some recite collections of verses related to the Korban Pesach and study some associated laws on 14 Nissan afternoon. This is a practice which significantly postdates the Talmud. Its existence enhances my question: clearly all those people think such a commemoration is necessary or at least reasonable. Why wasn't it legislated?)
Given the central value of this Korban and its being brought in such a communal fashion (Exodus 12:3-4, 12:47-49; Pesachim 64a, 65b, 76b, 91a; Rambam KP 1:3, 1:9-11; etc.), I would like to see more than "technically it's a Korban Yachid" in an answer.