But unto the place which the LORD your God shall choose out of all
your tribes to put his name there, even unto his habitation shall ye
seek, and thither thou shalt come: (Deuteronomy 12:5, KJV)
This command was given before the children of Israel had made their entrance into the promised land of Canaan. However, the people were already sacrificing in the tabernacle in the wilderness. The new place would be a place of worship that would be the central place for all Israel, but certainly it would not be the only place. For example, the Israelites were not all expected to gather in Jerusalem for worship every Sabbath--which would have been extremely difficult for many of them who lived farther away, if not impossible. There were several gatherings each year to which the men of Israel were commanded to assemble, and Jerusalem, which was the place later chosen by God for this, became the location for these festivals.
But look at the next two verses in Deuteronomy for appropriate context:
And thither ye shall bring your burnt offerings, and your sacrifices,
and your tithes, and heave offerings of your hand, and your vows, and
your freewill offerings, and the firstlings of your herds and of your
flocks: And there ye shall eat before the LORD your God, and ye shall
rejoice in all that ye put your hand unto, ye and your households,
wherein the LORD thy God hath blessed thee. (Deuteronomy 12:6-7, KJV)
Yes, to this central location they were to bring their tithes, their freewill offerings, their sacrifices, etc. But notice that it also says "And there ye shall eat before the LORD your God." If Jerusalem, and especially the Temple, were the only place for these activities, then it would follow that they should only eat there as well.
Clearly, Jerusalem was not the only place for worship. It was, however, the place chosen for the general assemblies.
Three times in a year shall all thy males appear before the LORD thy
God in the place which he shall choose; in the feast of unleavened
bread, and in the feast of weeks, and in the feast of tabernacles: and
they shall not appear before the LORD empty: (Deuteronomy 16:16, KJV)
The system of sacrifices did not end with the destruction of the Temple: they ended with the veil being rent at the death of Christ.
And the veil of the temple was rent in twain from the top to the
bottom. (Mark 15:38, KJV)
By the which will we are sanctified through the offering of the body
of Jesus Christ once for all. . . . By a new and living way, which he
hath consecrated for us, through the veil, that is to say, his flesh;
(Hebrews 10:10,20, KJV)
But their minds were blinded: for until this day remaineth the same
vail untaken away in the reading of the old testament; which vail is
done away in Christ. (2 Corinthians 3:14, KJV)
That veil being torn was God's signal that the Lamb of God had been offered, and that the system of sacrifices had forever ended. The Temple itself was not destroyed until 40 years later -- well after the sacrifices should have ceased. (The Jews, who did not recognize Jesus as the Messiah, did not acknowledge this signal and continued to uselessly offer sacrifices; but the Christians understood it and accepted Christ's sacrifice in their place.)
Conclusion
Because the command did not prohibit sacrifices in other places, the passage cannot be used as the sole support for the sacrificial system having ended with the destruction of the Temple.