The statement that Henry VII declared himself king because of his defeat of Richard III at the Battle of Bosworth needs a certain amount of contextualizing. He was not just some nobleman who thought to himself, "hmm, I'd like to be king and I think I could do it." He was, in fact, the nephew of Henry VI.

As I discussed in this previous answer about his claim, Henry VI's mother, Catherine of Valois, was Henry VII's grandmother. That did not, in and of itself, give him a claim to the throne, but it put him firmly into the orbit of the royal family and therefore the inner circle of the Lancastrian cause. In addition, he had royal blood on his mother's side as well - she was descended from (Prince) John of Gaunt, through one of his legitimized illegitimate children. However, this was problematic in and of itself.

You see, John had been married to Blanche of Lancaster and had children with her, including the future Henry IV. After her death he married Constance of Castile, and had a daughter, Catherine, who married Enrico III of Castile. At the same time, he had a long affair with Katherine Swynford and had these illegitimate children, who were given the surname Beaufort. John and Katherine married after Constance died, and the reigning king - Richard II, John's nephew - allowed their children to be legitimized by the Pope. However, Richard II was overthrown by his cousin Henry, and in order to secure his throne, Henry IV ruled that the Beauforts were not eligible to inherit it, or to pass a claim down to their descendants, which is what makes things complicated.

I argued in the previous answer that Henry VII's Beaufort ancestry was not the single deciding factor in his being perceived as the heir to the Lancastrian cause, and I still stand by that, but it was clearly a factor. If Henry VII's father hadn't been a Tudor half-brother to the king, he would probably not have had enough of a spotlight or been closely connected with enough powerful people to win the throne. On the other hand, without his mother's status as the only child of the eldest (surviving) son of the eldest Beaufort son, it's unlikely that he would have been able to present himself as a real contender, despite the legal bars to Beaufort inheritance.

But in any case, for that Wiki editor to write simply that he claimed the throne by right of conquest is an oversimplification. Henry actually had real trouble defining his ownership of the throne with clarity, and he asserted both his right by conquest and his right by blood to Parliament, and implicitly tied it up with his wife's inherited right to the throne through her father, Edward IV, to fully secure his children's rights. He couldn't have simply taken the throne by force of arms and then said, "this is mine because I took it" as the sole justification, because it would have emboldened anybody with no claim to the throne to try to take it from him or his descendants - and would have granted a huge propagandistic gift to the remaining Plantagenets, who did have a blood right and could have used the inherent admission of his lack of blood right to gain supporters. It was very important to prove that you had not only the strength to keep the throne (or rather, had God on your side to help you win battles, but that's a whole other question) but also the lineage to deserve it.

I also want to note that the monologue is not listing an "absurdly complicated line of descent" for Henry V to justify his right to the French throne - it's pointing out that this claim of "Salic Law" to justify disinheriting French princesses was actually relatively new, and that earlier French/Frankish kings had cared about the lineage of their mothers to prove their right to the throne. As I discussed in a previous answer on cadet lines,

The crown passed directly from father to son until we get to Philippe IV (1268-1314), who was succeeded by a son (Louis X), then an infant grandson who died soon after birth (Jean I), then another of his sons (Philippe V), and then another. This last son of Philippe IV, Charles IV, had only one child living - and she was a girl: Marie. His wife was pregnant at the time of his death, but the child turned out to be another girl, Blanche.

Unlike certain other countries, Capetian France never had a king before Louis X who didn't have a son to pass the crown to. His uncle Philippe was made regent until Louis's pregnant wife gave birth to Jean, and then decided that he should inherit the throne instead of Jean's older sister, Jeanne. He had the Estates-General declare that there was insufficient precedence for a woman to inherit, and Jeanne was allowed only the throne of Navarre. There was no problem with Charles's accession after Philippe's, but with no more Capet brothers to succeed him, it was necessary to take stronger steps. It was declared, on the basis of the old "Salic Law" - a law code written in the reign of Clovis I in the sixth century, which you may recognize as a really long time ago, so long ago that it's not really relevant by the fourteenth century - that women definitely had no right to a portion of the kingdom, no ability to inherit it or to pass it on to their children, which excluded Marie and Blanche.

Instead, Charles IV was succeeded by Philippe VI. Philippe IV's brother had been given titles by their royal father, the way non-heir royal children typically are, and one of them was "Count of Valois". This title was then inherited normally by his son, Philippe, and as a result, the branch of the family that inherited the throne became known as the Valois dynasty. Because it descends from a younger sibling of the main house, it is a cadet branch - that's all the term means.

Henry V's claim to the French throne went back to Edward III (father of John of Gaunt; his own great-grandfather), whose mother was Isabella of France, daughter of Philippe IV. When Charles IV died, Edward was willing to go along with the idea that women couldn't directly inherit, but felt that he had the greater right to the throne as the most closely related man in the family. Philippe VI was a more distant cousin of Philippe V and Charles IV, but it was declared that women couldn't even pass on a claim, and he was a male-line relation and, probably crucially, was able to act on it and take it immediately (much like Stephen of Blois usurping Empress Matilda). And Henry was not reaching back many generations - Edward had pressed this claim, and his sons would continue to as well. It was the central issue of the Hundred Years' War! They were continuously fighting about it. It would have been weirder for Henry V to go, "you know, I'm just going to keep fighting but I'm going to stop saying I have the right to this," than to make a point of his lineage.