If we are going to go down this road, then very well...

Being the descendant of Isildur does not, by birth right, make Aragorn king. When the two kingdoms of the Dunedin were formed in Middle Earth after the fall of Numenor, the North Kingdom of Arnor was ruled by the High King Elendil, and the South Kingdom of Gondor was ruled by his sons, Isildur and Anarion. [1][2]

During the first war of the ring against Sauron, Elendil and Anarion were both slain. After the war, Isildur, as the oldest surviving member of his line, became high king of the two kingdoms. However, when he left Gondor the year after the war and travelled north to Arnor, he installed his brother's son Meneldil as King of Gondor. On his journey north, Isildur was killed by a band of Orcs. The crown of Arnor was then passed to Isildur's only surviving son, Valandil. [3]

Valandil never made a claim to the thrown of Gondor. As a result the two kingdoms were separated. It was accepted in both kingdoms that Isildur, by installing Meneldil to the throne of Gondor, had forsaken his claim. From that point forward, Arnor was ruled by the line of Isildur and Gondor was ruled by the line of his brother Anarion. [1][2]

At one point in Gondor's history, the current king died without any obvious heir. At that time, Arvedui, king of Arnor, claimed the kingship of Gondor as the descendent of Isildur. Gondor refused the claim, saying that Isildur had granted the kingdom to his brother's line and only direct descendants of Anarion could be king. [4] Another later attempt at reunification was also rebuffed.

After the Second War of the Ring, where Sauron was finally destroyed, Aragorn again claimed lordship of both Arnor (which was destroyed but he hoped to rebuild) and Gondor. He was a direct descendant of Arvedui, who was rebuffed thousands of years before when he tried to reunite the kingdoms. [5] Throughout the Lord of the Rings, Aragorn is nervous about returning to Gondor because his claim is not assured.

The people of Gondor accept Aragorn's claim to the throne and he becomes the first High King of the Reunited Kingdom. [5] His lineage does play a part in their acceptance. However, the same lineage had been rebuffed several times in Gondor's past and was not seen as a mandate to rule.

Aragorn's actions during the war played a much larger role in the people accepting him as king. Much of this was due to an old prophecy, saying that "the hands of the king are the hands of a healer, and so shall the rightful king be known." When Aragorn healed the people wounded by the Black Breath of the Nazgul during the Battle of Pellenor Fields, the rumor went around the city that the king had returned. When he defeated Sauron, the people accepted him. [6]

So there you have it. Aragorn was considered a possible match for the Kingship due to being a descendant of Isildur. However, his lineage did not give him the right to claim the role on its own. The people of Gondor chose to disregard their ancient rules about the rights of succession in order to appoint him king. They did so because of his actions during the war and the prophecies he fulfilled. The main prophecy that planted the seed of him returning as king was his healing of the wounded in Minas Tirith. The article is more correct than you.

You know... if you want to be pedantic about it.

Sources:

1. http://tolkiengateway.net/wiki/Gondor

2. http://tolkiengateway.net/wiki/Arnor

3. http://tolkiengateway.net/wiki/Isildur

4. https://en.wikipedia.org/wiki/High-Kings_of_Arnor_and_Gondor

5. http://lotr.wikia.com/wiki/King_of_the_Reunited_Kingdom

6. http://tolkiengateway.net/wiki/Aragorn#Reign_as_Elessar

Edit: Fixed typo.