Try quoting more than the first half of a sentence intended to placate people who learned to believe in the fallacy.
the shortcomings of the theory had been recognized by the early 20th century, and it had been relegated to "biological mythology" by the mid-20th century
The Haeckelian form of recapitulation theory is considered defunct.
the Biogenetic Law was abandoned, and its fall freed scientists to appreciate the full range of embryonic changes that evolution can produce
AFAICT the comment you first responded to is consistent with the modern form of the idea. There's no reason to assume they were specifically thinking of Haeckellian recapitulation.
It is a fact that many embryos go through stages similar to portions of that organism's evolutionary history. This fact alone justifies the cautious phrasing of the great-grandparents comment, in my opinion. Since this fact is also indicated in the excerpted quote you criticize, I think the quote served it purpose. (Though I would agree the 'reflect' can be misinterpreted).
The cited book is 1963. I'd feel more comfortable with something more recent. It seems to be a debate of degree. Earlier forms do tend to be preserved in the embryo, but it's only a tendency, not a "rule".
the shortcomings of the theory had been recognized by the early 20th century, and it had been relegated to "biological mythology" by the mid-20th century
The Haeckelian form of recapitulation theory is considered defunct.
the Biogenetic Law was abandoned, and its fall freed scientists to appreciate the full range of embryonic changes that evolution can produce