May be you have already noticed too: The main story is the very example of illogicality.
The institute needs for whatever reason human DNA that is not contaminated by radiation.
They somehow get wind of Vault 111, packed full with cryogenically frozen pre-war humans, .... couples in fact.
Now guess what: Instead of 'rescuing' them all, spinning up a story of how the institute was the Vault-Tec backup for Vault maintenance or whatnot, convincing the dwellers of 111 to join the institute in union, which anyone under the given circumstance would (with a very high probability) readily agree to, yielding the institute with a limitless supply of pre-war DNA; the Institute instead only extracts an infant, killing its mother in the process and the leaves father as witness (or vice versa).... sounds like the plan to go doesn't it?
Even though anyone capable of the slightest hint of rational though might add one to one together and draw the conclusion that the surviving parent might not be the greatest fan of the organization that killed his husband/wife and abducted his/her son.
Again the logical way to go is of course, to grant that person (which may be fully armed and which may have allied itself with factions opposed to the Institute) unlimited access to the Institute.
The people living in the Institute are supposed to be scientists, who's daily bread and butter should be to anolyze and built hypotheses on given preconditions. How could such a bunch have come up with such a plan and I am using that term loosely?
Or are they suffering from some sort of brain-dysfunctionality due to the long isolation below ground in combination with inbreeding? - If so, how do they manage to accomplish everything else they accomplish?
Seriously Bethesda?