As far as I know, one cannot buy health insurance outside of open enrollment.
Yes, being dropped from a family plan feels like a life change event, but it is not according to the rules. This situation is like so many others when the government writes out rules and some people slip through the cracks- when it’s good for the person we think of it as a loophole, when it’s bad, we are all upset. The rules are written and no common sense interpretations can be applied, it is by the rules only.
I would not blame the doctors office, they only go by what the insurance company says. I had people who did sign up and were actively paying for exchange plans, but due to delays, the insurance companies saw them as not covered and took a month or two to update their insurance and then more time to redo the prior claims.
This is definitely a crack in the system. I would call several local agents and see if any of them have discovered any loopholes that could apply to this situation, they would only find this out in the way your friend found this problem. I am sure many people have not thought through and realized that this missing coverage issue exists. One thing might be possible, could they pay for add on maternity coverage via the work plan?