Poythress, in ch. 10 of the The Shadow of Christ in the Law of Moses, looks at Deuteronomy 13:1-18 and the conquest of Canaan in Joshua as primarily types for the Christian’s inner struggle against Satan and the spread of the gospel throughout the world. I think this overly spiritualizes the matter. I prefer to go where Poythress goes at the end of the chapter: to the Second Coming as the final fulfillment of what is anticipated in Deuteronomy and Joshua.
Thus, in Joshua, God is setting apart a pure land where his kingdom and people may exist as a beacon to the nations, to draw them to come and worship him. Therefore the land and the people had to be purified of false worship. The physical, land aspect of this is not unimportant; it ought not be presented as a type to spirituality in the NT/NC era. The real culmination of what happens in Joshua is the Second Coming of Christ. The tribulation and return of Christ to earth is a purification and conquest of the land (in this case, the whole earth) for the setting up of Christ’s kingdom on earth.
The reason the church does not carry out the death penalty for false worship or conduct holy wars against pagan nations is that it is not a national entity like Israel, and its purpose is not to establish a physical earthly kingdom. Rather, in God’s grace, a gap has appeared between the inauguration of Christ’s kingdom and its consummation in which all nations are invited and commanded to repent and enter the kingdom before the judgment arrives. Its role in God’s kingdom plan therefore differs from Israel’s and these Mosaic stipulations do not directly apply. They do apply as a warning against the greater wrath that is to come.