Videos
Every single Dumbledore we've seen (Albus, Aberforth, Ariana, and Credence) were all extremely powerful. They're extremely powerful in how much magical energy they can produce, dueling, inventions etc.
Dumbledore = For obvious reasons.
Aberforth (in the movies at least) = Produced arguably the largest patronus we've ever seen, combating multiple dementors at once. Just like Harry's incredibly rare feat in the forest. Given his immense dueling abilities, he was also able to hold his own in a 3-way duel with 2 of the most powerful wizards of all time, at age 15. This is something nobody would be able to boast about.
Ariana = Was hinted to be an Obscurial (which express their magic in such extreme ways) as a result of suppressing her magical abilities due to trauma. In contrast, Merope Gaunt's magic didn't turn her into an Obscurial when she repressed it due to trauma from her familiy. It made her magic weak/poor, to the point her father called her a squib. This strongly points to Obscurus cases only happening if the child was natural very powerful, whereas if their magical power was average or ordinary, it would be greatly weakened.
Credance aka Aurelius = A very powerful Obscurial who also used magic to fight Album in such a way we never saw anyone in the first 7 books/films use magic (expect the most powerful wizards in the HP universe). I mean, he would have been able to take on several death eaters in the Battle of Hogwarts and make quick work of the fight (based on what they showed us).
And why do they have a strong connection with phoenixes? Is it just unique to them? Is it like with the Slytherine bloodline that have a strong connection with snakes? Does this mean every wizarding bloodline have a special connection to a specific animal (magical or not)? Or is this just reserved for the most powerful bloodlines?
EDIT: Adding reasons why I think they're so powerful for each one.
You have asked for quotes from the book which have never actually addressed it, but in the film Fantastic Beasts: The Crimes of Grindelwald (written by J.K Rowling) it's confirmed that Credence Barebone is actually Aurelius Dumbledore.
In around 1901, Aberforth had a son, Aurelius Dumbledore with a girl that he fell in love with in Godric's Hollow. However, the girl was "sent away," presumably by her parents, either before or after Credence's birth, although Albus was aware of rumours that there was a child. Due to the strained relationship between the Dumbledore brothers, Aberforth neither confided in nor sought Albus' help. Aurelius was eventually sent away to America with an aunt, where he was adopted by Mary Lou Barebone and given a new name, Credence Barebone.
In 1932, Credence attacked Albus and accused him of abandoning him, revealing to Albus that he was supposedly a member of the Dumbledore family. During a duel between the two, Credence's phoenix spread ashes on his body and Albus realised that Credence really was a Dumbledore and that he was actually Aberforth's long-lost son. However, by this point, Credence was dying due to his nature as an Obscurial slowly poisoning him, something that Albus sadly recognised from his experiences with Ariana. After foiling Grindelwald's plans at the walk of the Qilin, Credence, in poor health, accepted his father's offer to come home with him.
Dumbledore Family Tree
So no, Albus and Aberforth were not the last Dumbledores. There was at least one other.
In the books, it’s easiest to infer that the Dumbledore line ended with Albus and Aberforth.
I think it is also difficult to say that the Dumbledores were an ancient wizarding family. After all, Percival married a Muggle-born so he can’t have been very pureblood leaning. If the extended Dumbledore family was, it is most likely they became extinct in the male line other than Percival.
Book 7, which is heavy on Dumbledore lore, speaks as if Albus’s nuclear family was the only Dumbledore family. We pretty much know Albus didn’t have kids so that’s the end of that line. As for Aberforth, I don’t think he’s the type to go around getting girls pregnant. Especially since it’s implied in the books he was a goatfu- uh, he preferred to devote his life on research and experiment on goats. I think from his lifestyle, he was probably asexual but that’s just me assuming.
In the FB movies, Aberforth has a kid so I guess that’s not the end of the line. I don’t know if Credence dies tho cuz I deleted those movies from my head. Nothing in those movies counts as canon information, even if Jo wrote it. Seeing as she changed McGonagall’s age by thirty to fifty years.