sste - agree with your post completely.
As several other posters have commented, it's not the fact that she quit at the 11th hour, but how she quit, especially after having communicated via phone, personal email, and in person with her future employer. It certainly could have been handled better in numerous other ways - a slightly lengthier explanation (one extra line - "due to unforeseen personal circumstances/emergency, I can't take the job"!), more notice, a phone call, meeting in person, offers to help find a replacement, or alerting the employer in advance that she might be looking for an alternative position - all of these would have mitigated the impression that she was behaving in a wholly unprofessional manner, and behavior that was unexpected based on her personal references.
Additionally, I think some discussion of this has blurred the line between babysitter and nanny - understandably, the definition may change depending on who you are speaking to. When the poster specifies that her prospective "nanny" has quit last minute, I'm assuming that this is a greater than part-time position, possibly salaried, paying employment/soc. sec. taxes and giving benefits. I say this employing a nanny currently, having participated in numerous threads on this board about nanny employment, and having numerous friends and colleagues who employ nannies. Admittedly there is a spectrum with regard to pay and benefits, but the majority of employers that I know and have communicated with, pay salaries not hourly wages, pay taxes on the nanny's salary, and give reasonable benefits (paid holidays off, paid vacation time, paid sick time). If a babysitter, who is a part-time, hourly wage employee, flakes on me last minute, I chalk it up to bad luck and move on. I'm much more invested in a nanny from the employment side and try to treat her as a professional - as such, I expect professionalism in return. This is where I assume the OP is coming from as well.
OP - I think you may have rubbed some people the wrong way with your initial comments about the nanny candidate's Christian character, and tying that into contacting the pastor. Leaving aside all discussion of character and religion, I still think it's reasonable (if you desire) to contact her most recent or main reference and give them an update along the lines of "I really appreciate your taking the time to recommend X for a nanny position in our family. I wanted to update you that although we offered her the position and she initially accepted, she ultimately turned the job down 12 hours prior to starting with no real explanation. I was surprised because it did not seem in keeping with your description of her professionalism and dependability and I ultimately decided that if I was her reference, I would want to be updated." They may think you're a crazy person. Or, it may be the most recent in a string of incidents that might make them think twice about vouching for this person. Either way, you've done what you felt necessary, and the situation is out of your hands.
And expecting DS1 10/2016