I think there are two distinct things going on in the quote (which is the first statement, on the first page, within the first paragraph of Quidditch through the Ages btw).
The second statement is about the transfigured witch or wizard (more like what happened to Draco in book four when Crouch/Moody transfigures him in the name of discipline). In the second example, the brain truly is reduced to that of the animal into which the witch or wizard is transfigured (however, Draco clearly understands that something happened to him and is frightened by the experience, which indicates to me there is some residual memory once one is transfigured back into his or her human shape).
However, as you also point out there may be more to the difficulties an animagus might face while in his/her animagus form as well.
First, we learn from Hermione in PoA, that the difference between an animagus and wereanimal is that in addition to the fact that the were-animal has no choice in transforming, the were-animal forgets himself entirely. This is further corroborated when Lupin says,
As long as I take it in the week preceding the full moon, I keep my
mind when I transform . . . I am able to curl up in my office, a
harmless wolf, and wait for the moon to wane again.
Before the wolfsbane potion was discovered, however, I became a fully
fledged monster once a month.
PoA pg. 353, Scholastic edition
As user 13267 points out, we learn from Sirius that Dementors were not able to detect him while he was in his dog form because his emotions at the very least become less human. Since McGonagall can still read maps and a paper when in her cat form (book one chapter one or two), clearly her intelligence was not reduced fully to animal form.
"Dementors can't see you know. . . " He swallowed. "They feel their
way toward people by feeding off their emotions. . . They could tell
that my feelings were less - less human, less complex when I was a
dog. . . but they thought, of course, that I was losing my mind like
everyone else in there, so it didn't trouble them. But I was weak,
very weak, and I had no hope of driving them away from me without a
wand. . . "
PoA pg. 371, Scholastic edition
With animagi it is very clear, memory and intelligence is retained both in the animagus form (why Sirius is able to get himself to Hogwarts to go after Pettigrew as well as why Rita Skeeter can spy as a beetle), but perhaps it is partially eclipsed by the emotions of the animal into which one transforms.
The witch/wizard and the animal form even have similarities when in the human shape - particularly during moments of high emotion. For example, if you look at the first few paragraphs of Chapter 18 in PoA, words used to describe the actions of Sirius include growled, teeth barred, snarled, and clawing. I'm quite sure it is no coincidence Rowling chose these words for Black at this moment having just revealed what he is. Pettigrew is also described as having a squeaky voice and she says "something of the rat lingered around his nose." So if those animal traits can impact the man even while in his human form, of course the animal emotions can impact the witch or wizard while in his animal form too and probably more strongly.
We also know from Prisoner of Azkaban (and the quote from Quidditch) that becoming an animagus is extremely difficult and perhaps even hazardous.
"There have only been seven animagi this century and Pettigrew's name
wasn't on the list."
Hermione - pg. 351
"it took the best part of three years to work out how to do it. Your
father and Sirius here were the cleverest students in the school, and
lucky they were, because the Animagus transformation can go horribly
wrong - one reason the ministry keeps a close watch on those
attempting to do it. Peter needed all the help he could get from
James and Sirius."
Lupin - pgs. 353-354 Scholastic edition
My understanding then, was that it certainly doesn't do long-term damage to intelligence, however, it may take someone with a very sound mind, and extreme focus to become an animagus in the first place. If it is rare to become any animal, it must be extremely rare to successfully become an animagus that also happens to be a flying animal.
Even then, it is clear that human emotions are diminished. This may put the animagus in danger of losing his/her human focus and desire and allow animal instict and emotion to have the potential of winning over (one way such a transformation might go horribly wrong and why, in part, the ministry keeps tabs on such things - or tries to).
So, we reach the first statement in the quote which indicates, as Yamikuronue points out in the comments, the quote simply points out the fact that flying animal animagi are rare. The quote also specifies flight in human form indicating flight can take place in the form of a flying animal - though this is difficult or rare.
No spell yet devised enables wizards to fly unaided in human form.
Those few Animagi who transform into winged creatures may enjoy
flight, but they are a rarity.
Considering Rita, there would actually have been 11 known animagi this century - in the British Isles. Only one of those, Rita, would have the ability to fly (most beetle species are fliers) - and I'm sure, fly she did as we see her very high up on a window ledge in order to eavesdrop.