Of course. I was focusing on the doing or recieving part and completely missed the second part: Are you a botherer and bothered OF/BY or ABOUT god?
It can be either one, so I'd say it depends on how religious and/or deranged the speaker is. Like you said - most would say it's about god, while I was deranged enough to interpret it as being a direct communication with a god.
It's crazy. My short stint working in child care I would always call the male parent first if I had their contacts, but it seems deeply ingrained in people to consider a mother "parent" and father "helper".
Maybe they'd be more comfortable calling you if you called them every day "just to check up, to see how the kid is doing", making it abundantly (and annoyingly) clear that you have nothing but time for your kid. And encourage your wife to respond "Have you called the primary contact?" whenever someone calls her.
I only have my own phone calendar, no shared alternatives unfortunately, but there is a multitude of options when you search for "family calendars". Or create a personal calendar for your child with both of you having the password and colour code it for who is responsible for each entry.