I am going to disagree with some of SM Stirling's characters although not necessarily with SM Stirling!
"'...are you sure this isn't our California, a long time ago?...And we could all go pop, like a soap bubble, if we change the things that made us.'"
-SM Stirling, Conquistador (New York, 2003), p. 26.
I discuss such issues on the Logic of Time Travel blog.
If X, born in timeline A, travels to timeline B and prevents the birth of X in timeline B, it follows that, in timeline B, X is not born, does not grow up and does not become a time traveler. However, it does not follow that the X who was born in timeline A, traveled to timeline B and prevented the birth of X in timeline B will cease to exist after preventing the birth of X in timeline B. There is no reason why X should cease to exist and there is a reason why he should continue to exist, namely the conservation laws.
Of course, scientific laws are empirical generalizations, not logical necessities. Thus, cessation of existence is logically possible at any time but there is no specific reason for the cessation of X's existence. What happens when a birth is prevented? Does the person who would have been born exist into adulthood and then cease to exist? No, he does not exist in the first place.
Christiansen thinks that time travel is self-contradictory (p.115). It is not but contradictory conclusions are often drawn from it. An effect (a time traveler's arrival) preceding its cause (the time traveler's departure) is not contradictory. However, an effect followed within a single timeline both by departure and by prevention of departure is contradictory. Several consistent scenarios allow for time travel. Poul Anderson's Time Patrol is a borderline case.