Blinking Out Of Space-Time?

Poul Anderson, Time Patrol (New York, 2006).

Manson Everard in a divergent timeline thinks:

"There were other Time Patrolmen in the pre-Roman past. They'd return to their respective eras and...
"Everard stiffened. A chill ran down his back and congealed in his belly.
"They'd return and see what had happened, and try to correct the trouble. If any of them succeeded, this world would blink out of space-time, and he would go with it." (p. 204)

When I first read that over fifty years ago, I thought, "The longer Everard remains in the divergent timeline, the greater the risk that he will be in it when it blinks out of existence," but, of course, this "...when..." does not refer to any moment within that timeline. At the moment when Everard thinks this, either he exists or he does not. The fact that he is thinking anything proves that he exists. Therefore, it is not true that he does not exist. Period. There are other points of view from which Everard no longer exists, for example the point of view of someone thinking about Everard after his death or the point of view of someone in a subsequent timeline but none of that changes the fact that Everard does exist now.

When Everard and Van Sarawak, accompanied by Deirdre, escape on a timecycle and take stock of where/when they have escaped to, Van Sarawak says:

"'What year is this? About the time of Christ? Then we're still upstairs of the turning point.'" (p. 217)

I thought, "Then travel further back without delay in case you are still in the divergent timeline when it blinks out of existence," but that was nonsense. The problem is that, when fictitious characters travel from, in this case, 1960 AD to about 30 AD, we imagine that 1960 and 30 and the date of the Second Punic War a few centuries BC are not different times but different places co-existing at the same time. Thus, we think, if a Time Patrolman arrives in say 213 BC and spends half an hour in that year before taking the action necessary to restore Roman victory in that war, then, half an hour after that Time Patrolman's arrival in 213 BC, the 30 AD and 1960 AD in which the Carthaginians had won that war will blink out of existence and, if Everard has lingered for more than half an hour in either of those years, then he will blink out of existence with it.

I hope that this clarification of intuitive thoughts demonstrates their absurdity.