I am recently divorced and have 2 dc, age 14 and 9, both very sensible, mature children who get on well. We live in a safe area and have nice neighbours we know well and who are around during the day.
I left the DC alone together in the house for an hour one morning so I could go on an errand (tmi but didn't think they'd want to accompany me for my smear test!). Ex-H found out in conversation and was cross. Doesn't mind older one being left but "won't let" me leave the younger one at home with her sister. I wouldn't leave the younger one on her own anyway. I wouldn't leave the older one to babysit in the evenings yet either.
I think he's being overprotective, both girls have phones, common sense and said they are happy to be left. In fact the youngest thinks it most unfair that she can't be left, so both ended up coming out today on the dullest of errands, which no one enjoyed much.
This is the first time post separation we've really disagreed, and i don't know what to do - parent together or separately. He has help from his parents when dc are with him so this probably hasn't been a practical issue for him.