They've brought it up in a few discussions regarding AMA in the past and it's a "best practices" approach, not a strict requirement for any and all AMA protocols. The reasons are primarily common sense... If you really don't want the pt to leave, then trying to obtain an ally in the family member/friend to help dissuade the pt from leaving is probably more effective than any further efforts by you alone. It also helps support your overall intent to do "any and everything possible" to prevent the pt from harming themselves by ill advised departure from the ER.
I agree that it's illogical to expect you to call a family member or friend unless the pt is asking you to talk to their wife, friend, etc.. about the disposition. However, if they are present in the room, I think trying to gain their support in persuading the pt to stay, documenting their name/relation, and/or obtaining their additional signature on the AMA form is prudent. It lessens any confusion and makes it more difficult for the pt and/or family member to argue that they didn't understand, or argue that you didn't try to talk them out of it, etc.. In case of a bad outcome, it's less likely that a family member would pursue litigation when they distinctly remember you having a conversation with them about attempts to dissuade their family member from leaving the ER, and also requested their signature on the AMA form, etc..
Is all of that necessary for you to successfully defend a related suit? Probably not, but it certainly makes it easier.
I always document things like "strongly recommended admission and/or further testing", "explained risks in leaving such as but not limited to death/perm disability". I document "why" the pt wanted to leave. I document the the pt verbalized understanding and demonstrated medical decision making capacity. I document that I attempted to engage family and/or friends to dissuade the pt from leaving but those efforts were unsuccessful. (I mention names and relationship.) I document that the pt was of sound mind and elected to take his chances in leaving. I obtain AMA signatures if possible. Most importantly....I make sure the nurse is in the room or I inform her immediately of the results of the discussion so she can document.
All of this sounds silly when you're faced with a homeless, belligerent, hostile, angry patient, spitting on the floor and cussing you out. Most of the times, it's not those guys you need to worry about. It's the family that comes after you for a bad outcome and/or death.
But I do agree with you, I wouldn't call anyone either in the example that you mentioned.