While the rest of the quote is completely dishonest I have a little something to say about this:
Another reason to reject their reported ages is that they dismiss written records detailing Britain’s past and similar records from several ancient European nations that trace royal ancestries all the way back to Japheth son of Noah.
While it is true that such documents exist, there is no reason to think that they are anything more than Christian attempts to integrate the pagan histories - such as they were - with Christian belief.
One of the cited sources for this claim is Snorri Sturluson's Edda. Sturluson was an Icelandic historian who tried to preserve the folklore of his people. If the connection was genuine Sturluson would be more likely to know of it than the monks who wrote the other documents - and as a Christian he had no motive to deny it.
However, Sturluson is not cited to support the connection with Noah - the material cited stops short, naming only a figure from pagan folklore identified with Japeth - by some. On checking I found the reason for the omission. Sturluson had also integrated Christian belief with the folklore histories - but he placed the Flood much further back in time, contradicting the claim. Needless to say, I found this omission more than a little dishonest. Admitting and answering this evidence would be the only honest way to handle it. Ignoring it and hoping that nobody notices is neither honest, nor sensible.
As a side note I have also seen Creationist's cite the Edda as evidence of the Flood in Norse mythology. In fact - as is perfectly obvious to anyone who reads it - the Flood story is taken from the Bible.