How do dispensationalists understand the statements that God fulfilled all his promises to Israel in Joshua 21 and 23?

Joshua 22:15-19 relativizes what is being interpreted in chapters 21 and 23. It makes a clear distinction between the Lord's Land and the land east of the Jordan, Gilead.

They came to the descendants of Reuben, Gad, and the half-tribe of Manasseh, to the land of Gilead [...] cross over to the land of Yahweh’s property [West, Canaan], where Yahweh’s tabernacle resides, and take possession among us.

Numbers 32 also tells us the story of those Transjordan tribes (Gilead), how Moses became angry when they wanted to stay East of the Jordan and how he only allowed it if their men came to fight in the West (Canaan). There was a clear opposition (from God and Moses) to fill the eastern portion of the Promise and the lands East of the Jordan (Moab et al.) always played a secondary role in Bible history.

The area between the Nile and the Euphrates (Arabian Peninsula) is also the logical candidate for the location of the New Jerusalem, fulfilling in this way perfectly the promise of the land. Zec 14:8 tells us that the New Jerusalem will connect the 'western sea' (certainly Mediterranean Sea) and the 'eastern sea' (probably Arabian Sea). And it is surely remarkable (but regularly neglected because the numbers in Revelation are constantly watered down) that the length of the Red Sea and two of the borders of the Arabian Peninsula almost precisely fit the dimensions provided in Revelation 21 (12000 stadia = 2200kms", Rev 21:9-14). With all those indications (plus the mountain of God (Zion / Mt. Sinai in Saudi Arabia), we can hardly assume that the New Jerusalem will be located in any part of the world other than the area of, and immediately adjacent to current Israel.

We have to understand that Canaan is only a precursor of the Promised Land. A shadow of the real Promised Land what is yet about to come.

I've read the article you quoted from Chosen People Ministries, and I noticed the writer(s) of the article assume the river mentioned in God's promise to Abram in Genesis 15 is the Nile. Not all Bible scholars, however, believe the river God mentioned to Abram was the Nile. A footnote in the NET Bible identifies the river as a wadi, not a river:

sn The river of Egypt is a wadi (a seasonal stream) on the northeastern border of Egypt, not the River Nile.

If in fact the river God spoke of to Abram was a wadi, not a river, then the land God ceded to Abram's descendants did not include a significant amount of land that the folks at Chosen People Ministry claim was included in God's promise.

Consider the following answer, provided by the website "Got Questions". Notice I've provided the biblical text in a paragraph all its own:

Some point to a passage later in the book of Joshua as contradicting the promise of Joshua 1:4. After the conquest of Canaan, the historical account says,

So the LORD gave Israel all the land which He had sworn to give to their fathers, and they possessed it and lived in it. And the LORD gave them rest on every side, according to all that He had sworn to their fathers, and no one of all their enemies stood before them; the LORD gave all their enemies into their hand. Not one of the good promises which the LORD had made to the house of Israel failed; all came to pass” (Joshua 21:43–45).

There is really no contradiction. At the time referred to in Joshua 21, all of Israel’s enemies were subdued. No one posed a threat to God’s people. God had given them a right to everything He had promised in Joshua 1:4, and they were authorized to take possession of the entire land—all the way to the Euphrates—as soon as they needed it and as soon as they called on the Lord for aid. The fact that they never did so does not negate the fact that God had kept His promise.

Since you mentioned Dispensational theology in your question, allow me in conclusion to defend Dispensationalism, which had its roots in the Plymouth Brethren Movement that began in Dublin, Ireland, with only a handful of people, including Dr. Edward Cronin:

Dr. Edward Cronin was saved while a student in Dublin. He was happily received as a visitor at a number of protestant churches, but when it became clear he would be remaining in Dublin, he was urged to apply for “Special Membership” in one of them. Finding the idea of “membership” in a local church to be distasteful, he withdrew from these churches. In 1825, together with his two cousins (the Misses Drury) and Mr. Tims, he began to meet in his house in Lower Pembroke Street. By 1827, this meeting had grown so much that Mr. H. Hutchinson offered the use of a larger room at 9 Fitzwilliam Street. By 1830, they were using a large auction room at 11 Aungier Street. Given that this movement started in Dublin, the inaccuracy of the title “Plymouth Brethren” should be clear.

The first meetings in Dublin were held in 1825. By 1827, J.N. Darby and J.G. Bellett were attending the meeting. By 1830, there were five or six [Brethren] meetings in Ireland. At the same time, Christians in England were becoming exercised about the Scriptural principles the believers in Dublin were rediscovering. It was also in 1830 that the first meeting in London England started. Darby went to Plymouth [hence the origin of what was originally an epithet(!), > "those Plymouth Brethren"] in 1832 at the request of Benjamin Wills Newton . . .. This meeting in Plymouth was initially at Ebrington Street, and [it] was later transferred to Compton Street. By 1840, the Plymouth meeting had grown to 800 persons, and by 1845 to 1200 persons. Mr. H. Hutchinson offered the use of a larger room at 9 Fitzwilliam Street. By 1830, they were using a large auction room at 11 Aungier Street.

I thank God for my upbringing in the Plymouth Brethren (aka the PBs), and for my exposure to the Dispensational tradition. To this day--at age 71--I am convinced that Dispensationalism is but one of many biblical paradigms that have solid biblical backing.

"We are rightly dividing the Word of truth" (2 Timothy 2:15 KJV) is a claim made by any number of denominations, not just the Plymouth Brethren, and the claim is very much like a paradigm through which a denomination interprets the Bible writ large. Is each one of those paradigms "the truth, the whole truth, and nothing but the truth"? I don't think so. Each paradigm, I suppose, has its flaws.

I suggest the only person who benefits from the various schisms that occurred in the wake of disagreements about whose paradigm is best is the archenemy of our souls, Satan. Are some paradigms clearly wrong? Yes. Are some paradigms missing some key--and clearly biblical--elements? Yes. I also suggest, however, that each paradigm has something good to offer the diligent Bible scholar and every student of the Bible who may not be a scholar.

For any denomination to refuse to engage in a citywide evangelistic effort, for example, on the basis of another denominations's paradigm being different from theirs, is a clear repudiation of the unity that should exist within the Body of Christ. Enough said.