For those few times that we did want our daughter to go into the lounge post 11 AM unaccompanied (last time we tried this was over 3 years ago), I asked the concierge to accompany her to the lounge. She did accompany her to the food area, and then she was brought by the concierge to some private AC area, to consume the food and beverages.

About the straddling of the 11 AM issue, I believe the lounge agents looked at the time of the departing flight to decide about eligibility to enter the lounge, or not. So with your example, my daughter would not have been allowed into the lounge.