In Deuteronomy 12:21 G-d tells the Jewish people that when they arrive to their homeland and settle into the areas they will be assigned, "[t]hen you may slaughter of your herd and flock...as I have commanded you." Apparently this is a reference to a method of slaughter. Where is that commandment written down?
The fifth book of Moses is a "recap" or resummation, which is why the book is the "deuteronomy" of the Law of Moses. The following discussion will explain that there are no new (or missing) commandments.
If one looks at the first four verses of Deuteronomy, one can see that Moses compiled this fifth book during the last few weeks of the fortieth year of the wilderness wanderings.
In other words, Moses provided a "recap" of the law given at Sinai, since those Israelites entering the Promised Land were the second generation from those who escaped Egypt (who of course were the first generation).
One of the darker chapters of the Jewish people is the fact that the 99.8% of the adult population of Israelites who escaped Egypt with Moses were not "saved." They wandered the wilderness "until their bodies dropped (dead)" according to Hebrews 3:17 (which references Psalm 95:8-11). These first generation Israelites disassociated themselves from the Abrahamic Covenant as they refused to circumcise the second generation of male children, who subsequently entered the Land of Promise uncircumcised (Joshua 5:4-5). So when Moses formulated his fifth book, he was providing this next generation of Israelites a "recap" of the Mosaic Law right before they entered the Promised Land.
Thus this fifth book of Moses is called in English "Deuteronomy" because this fifth book of Moses is a "recap" of the Mosaic Law that was originally provided to those who escaped Egypt. It ends with a song that was supposed to be memorized by the Israelites (Deut 32:1-43). This song captured what happened to the original generation of Israelites who escaped Egypt and the curses they brought upon themselves (not to mention any other future generation that also disregarded the same Mosaic Law).
So Deuteronomy 12 is a "recap" of Leviticus 17. There is no missing commandment, because Deuteronomy was written weeks or months before the Israelites entered the Promised Land (and therefore was written chronologically AFTER Leviticus had already been given). Therefore Deuteronomy 12 was mentioning what had already been revealed in Leviticus 17. That is, the methodology of the blood sacrifice process (Lev 17) not to mention the requirement for tithing to/through the priests (Lev 27) and celebrating the feasts geographically in the same location of the presence of the Lord (Lev 23). Thus there are no "new" (or missing) commandments in Deuteronomy 12, although Moses of course amplified several of God's commandments throughout the Book of Deuteronomy to the second generation of Israelites in the context of their immanent entrance and life inside the Promised Land.
So the principal purpose of the Fifth Book of Moses was to provide this second generation of Israelites a sacred "pep talk" which comprised a "recap" of the Covenant at Sinai (and its attendant requirements for all future generations) -- thus the Fifth Book of Moses in English is called "Deutero"nomy.
To quote Rashi, which you already know:
But this seems strange to me: Deuteronomy 12:20-28 seems to be a restatement of Deuteronomy 15-19. In particular, verse 21 seems to refer back to verses 15-16:
In the context of the chapter, offerings to the Lord must be made in the place where God chooses once the people of Israel settle across the Jordan. (That seems to have been Shiloh in the period of Joshua and Judges, but Jerusalem after David became king.) Altars to Canaanite gods were to be destroyed (see verses 2-4). Ritual killing of animals should not be done in the places where non-Israelites had sacrificed to their gods.
On the other hand, ordinary slaughter of animals could be done outside of the centralized place of worship. There's no particular restriction as long as the blood is drained from the animal, which is the restriction imposed on Noah and his descendants in Genesis 9. (For some reason, Rashi says that verse 12 refers to animals that were originally designated for sacrifice and later became blemished. I don't see any evidence for that in the text.)
Why two statements of the same thing?
But this interpretation leaves a small puzzle: why the repetition? Note that the first statement expresses a concern for the Levites, who do not receive an inheritance, but are given the fruit of the tithe. The concern seems to be that if people slaughter their animals at home instead of going to the temple, the Levites will not be provided for. So the first statement of the allowance ends with an admonition to not neglect the Levite.
The second statement begins with an acknowledgment that when the territory of Israel expands, some people will live further away from the sacrificial center than others. Rather than give up eating meat, those people should feel free to butcher animals where ever they live as long as they continue to follow the regulations that God has already given them. In this case, the concern is for the people who live in distant towns.