I mean, your criticisms don't make a lot of sense here.
Firstly, we don't know that anyone did use protection. They may well just not have, because it's trivial to detect and fix whilst the embryo is just a few cells. Or she intentionally didn't. So there's no real criticism there. If they'd said something about using protection and it failing then that discussion would make sense.
Secondly, people in Picard's era routinely live to 120+ years old. In TNG, when McCoy appeared, he looked to be in his 90s, and he was 137 years old, and still footling around under his own power!
Also buddy, I know you're a dude and probably don't know a lot about pregnancy, but people have become pregnant in their 50s even IRL - the oldest natural (non-IVF) pregancy was a woman who was 59. That may well be with HRT etc. but I'm sure Bev would be getting anything she needed to keep rollin' there. If people are living to 120+ routinely, it may well be more like 60-80 is the normal cut-off.
The real question is actually why does Picard look so old when he's "only" 96.