I don't see how the presence of a few hundred "time aliens" at any given year could be all that disruptive. Obviously, the Star Trek time travel logic doesn't allow for a "butterfly effect" where small disruptions lead to major alterations in outcome, even though logically that should be the case: if Captain Kirk goes back to 1234 AD and takes a deep breath, that will most probably prevent my birth which is the sum of many coincidences and split-second timings, yet in the Trek universe something damps out such effects. This something should be enough to damp out the involuntary alterations brought about by those hundred visitors, too, especially if they all merely wish to live out their lives without disturbing anybody.
That assumes that the entire population of billions did relocate into the past, of course. It's equally possible, if not more so, that the majority of the population did not - that they either actively committed suicide, or then merely decided not to have children in the remaining century or so. Just a small handful of people might have remained during those final few years to escape to the past.
Of course, even if people going to the past changed it radically, would we know? The final outcome would probably still be the same: sometime during the final century or so, these people would learn of the impending doom and would build the time machine, quite regardless of whether preceding "cycles" had done the same and massively disrupted the past. Indeed, the dictator Zor Khan's reign might be one such case of a time traveler taking advantage of his special knowledge and grabbing power: the fact that Zor used time travel as a means of punishment suggests that he had indeed mastered the use and consequences of the technology and was confident that there would be no backlash...
I don't see why the Library or the Librarian would have to disappear once all the remaining folks had been "evacuated". To the contrary, both the Library and the Librarian would be the work of a generation that had preceded the exodus, and this past generation would not disappear merely because their children decided to step into a time machine; nor would its achievements.
In the episode, because our heros were not 'prepared', as Atoz wanted, they would die.
This was discussed recently. It seems that our heroes may have been rather mistaken about this "preparation" thing and its effects - and that Mr. Atoz may have been deliberately misleading his victims with his mumbo-jumbo about preparation, a procedure that might merely involve mental reconditioning that prevents homesickness.
Timo Saloniemi