Is the information contained in this book outdated? (Just as one example, there's quite a lot of passages on phonology, a field I can imagine have progressed quite heavily since 1906 when the book was written, even if the information in the book might be of a more basic nature and therefore haven't changed.)
Disclaimer: I did not read these books.
But from my own study of singing I think the biggest problem will be style(performance) not the medicine or method. From the oldest recordings we have we can tell that even less time ago the singing style is not preferred by todays standards. But maybe in 20 years it is again.
So yes, I would expect outdated, but not as in "not working" or even unhealthy. Just not the trend. The human from ~100 years ago is exactly the same human as today, so what worked then for your voice works now.
A rule of thumb: If there are lessons and units in that book, just do them. Follow the strategy of the book. If there is talk about anatomy and medicine, read them but don't trust them. 1906 may not sound far away, in music terms, but in medicine it really it. And the important part: It is only text in these books, no teacher from that time and no Audio-CD to imitate :).
Technical lessons to connect your lower register with your middle register (for example) are still perfectly valid. There is a good chance that the same lessons are still done today somewhere (on professional/university level).
Point is: Don't worry, the style is made by your ears and by trying to reach a certain sound you may already aim at, not dictated by your practice method.
Any book which goes basically like "try to sing these notes in the following manner to achieve the following goal" will work.