Let me begin with a couple of technical points. I believe the New Testament verse to which you refer is Matthew 3:12. In the King James Bible, it refers simply to the "floor" but this is obviously a threshingfloor, so there is no problem. Also, the same phrase is used in Luke 3:17, so there are two New Testament references to the threshingfloor.
Threshing in Bible times was the harvest-time activity by which the grain was removed from the husk and the tares (or false grain) was separated from the wheat. The threshing floor was the place where this activity took place and it was an important place for the people of Bible times much like the importance of the water-powered mill in my part of the world about 150 years ago. The threshing floor was flat and hardened by the passing of oxen over the sheaves, sometimes pulling a sled designed to separate the grain from the husks. After the threshing process, the stalks and grain were thrown up into the air so that the wind might blow the unwanted chaff away and leave the valuable kernels. This was known as winnowing. The threshing floors were important places and were often used as landmarks.
As I read through the Bible uses of the threshing floor, it clearly has two connotations. First, it is a place of blessing. It was the place where the grain of the harvest was actually taken from the sheaves. As such, it was a place where the blessing was received. The Bible refers to "the increase of the threshingfloor" (Numbers 18:30) and to a time when "the floors shall be full of wheat" (Joel 2:24).
But secondly, it is a picture of judgment. This is certainly seen in the Matthew 3:12 passage you mentioned: "Whose fan is in his hand, and he will throughly purge his floor, and gather his wheat into the garner; but he will burn up the chaff with unquenchable fire." At the time of threshing, the wheat is separated from the chaff and the tares. Also, the sheaves are beaten or crushed in order to make the separation. All of this is seen as a picture of judgment. Micah 4:12 tells how the Lord will gather the nations "as the sheaves into the floor." The picture of judgment is very clear.
So, why did God allow David to purchase the threshing floor from the Jebusite and use it as the temple site? Of course, it is the location of Mount Moriah and the offering of Isaac. 2 Chronicles 3:1 states, "Then Solomon began to build the house of the LORD at Jerusalem in mount Moriah, where the LORD appeared unto David his father, in the place that David had prepared in the threshingfloor of Ornan the Jebusite." But how does the typical significance of a threshing floor fit with the temple?
First, it is a place of blessing. Certainly, this can be said of the place where God choose to meet with the Israelites. In the Psalms, we read, "Blessed are they that dwell in thy house" (Psalm 84:4) and "we have blessed you out of the house of the LORD" (Psalm 118:26). Other passages point to the same truth. The temple was meant to be a place of blessing for the people of God.
Second, it is a place of judgment. We see this in the initial purchase of the site by David. God was judging Israel because of David's sin and David needed to bring an offering to the Lord. But we also see this in the brazen altar in front of the temple. It was a place of sacrifice where the sins of the Israelites would be imputed to the sacrificial animal and forgiven.
I am sure that much more could be drawn from this. I encourage you to take this study further.