The source for the Mitzvah that everyone is responsible for burying a Mes Mitzvah is unclear. The Mitzvah that relatives are responsible to bury each other also not explicit in the Torah; we learn it from the Mitzvah of a Kohen to be Metamei to relatives, and that Mitzvah is then applied to Yisraelim as well (according to many Rishonim; however the Rambam seems to rule in Hilchos Avel 2:6 that Yisraelim are not commanded by the Torah to bury their relatives; see Insights to Kesuvos 95:2:b).

Nevertheless, Kevurah, burial under the ground, itself is certainly a Mitzvah mid'Oraisa (see the Mishnah at the beginning of the third Perek of Berachos, which states that those who go to bury the dead are Patur from Keri'as Shema since they are Osek b'Mitzvah). The Gemara in Sanhedrin 46b learns this Mitzvah from the words "Kavor Tivberenu," and the Yerushalmi (Kil'ayim 9:3) learns this Mitzvah from "El Afar Tashuv" (see TORAS HA'ADAM, Inyan ha'Kevurah, who discusses this in more detail; see also RAMBAN to Bereishis 23:8, and Rav Chavell's notes there from the Kli Chemdah). In addition, it is considered to be a great disgrace for mankind in general to leave a person unburied (Sanhedrin, ibid.).

This appears to be the source the Rabanan saw for attaching such importance to the Kevurah of a Mes Mitzvah; the importance of Kavod ha'Beriyos (see especially Berachos 20b). Therefore Chazal saw to learn from this verse that the exception to the laws of Isur Tum'ah must be a Mes Mitzvah, rather than anything else. (This is not because the "Aseh" of Mes Mitzvah pushes aside other Mitzvos, as you imply. Rather, it is a Halachah that is learned from the verse "l'Aviv ul'Imo"; see Tosfos in Berachos 20a as to why this is different from other cases of "Aseh Docheh lo Ta'aseh.")

Be well,

Dov Zupnik and M. Kornfeld