It was never the problem many feared it would be. That’s not to say it could not have been much worse.
Birth records were manageable, as has been noted, at least by sentient beings. I have never heard of a computer that was self-aware.
The fears were real, certainly:
Russian defensive systems? Say there are a whole lot of missiles somewhere, all fitted with powerful warheads, ready for activation following command & control protocols. But somewhere along the line a bunch of not-so-bright people added some computer code, maybe for the purpose of enforcing a maintenance regimen, okay? Things like that typically also have “fail safe” mechanisms. So here are these missiles, and their automatic C&C system detects a failure condition (like maybe the missiles haven’t been maintained in an awfully long time). The problem is that “failure” can be handled in many ways. For example, the things simply power themselves down, or make themselves impossible to launch; those would have been proper responses. But suppose these missiles had been emplaced by a different government regime, such as the Soviet Union… What if the Soviet response to a failure condition had been to interpret it as proof of a past attack by NATO… And what if the coded response to a failure was to immediately launch all missiles toward predetermined targets? That was a very real fear among a lot of our DoD “smart people”.
Our at least fairly well-developed “limited ballistic missile defense”? See above, but change the actors around appropriately.
I was a systems analyst at the time, and there was a very real fear that an awful lot of things like computer code revision control systems would fail due to underflow/overflow/division-by-zero faults if a date field that wasn’t (as expected) iteratively and linearly progressing in a monotonically increasing direction–“99” suddenly became “00” instead of “100”–in such a case chaos might ensue, at least for a while.
In June, 1998, my government bosses determined that, if the ~500 computers I managed at the time were not all either patched to address these kinds of problems (or, if they could not be patched, they were not taken out of service) before October 1, 1998, the large battle lab where I worked would be taken completely off line, the power cut, and the doors locked. It was a rough few months!