Two Different Tabernacles It is true that the Mosaic Tabernacle remained in Gibeon, and along with it the Mosaic Altar, the High Priesthood, and importantly, the Ephod. What was missing was the "Ark of the Covenant.
David set up a Tent in Jerusalem to house the recovered Ark of the Covenant. But it was not an elaborate tent made of badger skins, purple tarp, etc., like the Mosaic Tabernacle.
However, what it did have was great musical worship! Worship accompanied with instruments invented by Kind David, also known as the sweet psalmist of Israel. David enhanced worship by writing exquisite psalms that exulted the glory of Jehovah.
Typology? When the problem of assimilating Gentiles into the Early Church arose, the Apostles appealed to the erection of this Davidic tent, which was mentioned in an Old Testament prophecy by Amos:
After these things I will return, and I will rebuild the fallen booth of David.
I will rebuild its ruins, and I will restore it, so that the rest of mankind may seek the LORD, and all the Gentiles who are called by My Name, says the LORD, who makes these things known from long ago. (Acts 15:15-18; the Apostles using the LXX version. LSB)
Although the Mosaic Tabernacle's typology is appealed to by other N.T. writers (especially Hebrews) with great meaning to illustrate the realities of the Gospel, this booth of David is mentioned and used in the instructive letter to the Gentiles because it would be more relatable for the Gentiles who were not familiar with Jewish religious customs and requirements.
Also several commentators consider the type of worship in this Booth (Tent) to be a significant point worth considering, that they deem important for the reason of the success in establishing the new Church throughout the world. Exuberant singing by, not just the ministers, but by everyone from Royalty (like David) to the common man (like you and me!). In the Presence of God!
It is a worship emulated today, unfettered by rules and regulations, that brings disciples into the holy atmosphere of the Divine.