I have travelled many times with my kids without DH, only ever to the US. Most of the time driving across the border(flying from BUF) but sometimes flying from YYZ. I always had a letter that was written on DH's company letterhead, signed by both of us and signed by his boss along with everyone's business cards attached. The letter included DH's consent/knowledge of the trip, all our passport info (incl DH's), all dates, airline and hotel info, DH's consent for me to make any medical decisions, all our phone numbers and address. Then a sentence or two from DH's boss stating who he was and confirmed all info.....I guess a witness of sorts. My kids were prepared to speak for themselves should they be asked by border agents. They were questioned maybe 50% of the time. I would also say I was asked for the letter 50% of the time. The first time I took my youngest away she was 7 and her passport had not been signed....that was the only thing that was questioned. The agent asked her if she could write her name, she said yes and he handed her a pen.
My kids also travelled with my mother and those trips we got the letter notarized. She never took them to the US...they went to Mexico. Those times the kids were never asked anything but my mother had to produce the letter. Oddly, the one time my mother was not asked for the letter in Canada at airport check in, she was asked in MX when checking in for the return flight.