Probably it is just embarrassment...we don't usually 'talk' about some subjects with family members, anyway...even the smaller issues.
We expect that they understand, from their own experiences, and respect our need for privacy as we mature.
I think it is lovely that her dad offered. What confidence that instills in the little one...that she CAN talk to her dad if the pain gets too big for her to handle.
...In the book called 'Journey', the mother of a hemopheliac (sp?) son tells that she had to sit with her boy while he screamed in agony as his non-clotting blood filled any injured joint, until the pressure of the blood-filled cavity, itself, stopped the bleeding, and he fell asleep in exhaustion.
That was the only remedy they had for that type of injury, for hemopheliacs, in those days.
The mother understood this, but that didn't stop HER pain.
But, she had a friend (named Margaret, I think), who told the mother to please call her, at any time, day or night, if she just needed to talk to someone.
The mother never did call, but she tells of the many times she stood by the telephone, grateful for the knowledge that she COULD call, if her pain got the better of her.
I thought that was a dandy idea. Margaret had given the mother an escape, a listening, caring, and trustworthy ear...just in case it was needed.
I passed on that idea, availability, to a number of people, over the decades since I read it. I remember that one friend called with the news of her husband's passing, at 2 a.m. I was so glad she did. I remember standing by my kitchen sink, in the dark, and in awe of this lady's composure...still she felt free to call at that hour, when she wanted to. I felt honored.
(Nor did I know much to say. But, it comes to you.)
It is such an easy gift to give...openness.
I know that counselors of all types must be guided by 'rules', just in case they don't yet know all the in's and out's...but I suspect that you are rapidly becoming 'experienced', and able to call upon your great good sense.
That's a 'gift', you know...inborn. Only life experiences can reveal it.
I think your young friend has it, too...that understanding that she, her inner self, is above and beyond any 'event'.
You say, "Normally the sessions are in groups, but this is something best kept private."
In my seven decades I've seen some very silly things done with misplaced information. The recipient misses the opportunity to just 'be there' for the speaker...just listening...and keeping shut...a 'sounding board', if you will.
A famous author, Barbara Sher, describes a 'pity party', a time-limited rant, where the listeners are NOT allowed to give any suggestions, only to cheer the ranter on. (This is when the 'victim' feels they need to get their troubles out.)
The safe listeners may shout stuff like, "Oh, that was a good one," and, "You tell 'em, Rosalie!"
This continues for only 15 minutes, or until everyone dissolves in laughter before that (if laughter is possible and appropriate).
I think this 'exercise' is in the free ebook, her first best-seller, "Wishcraft"...
Perhaps it is only necessary to have a 'pity party' as an option, just as the possibility of a phone call.
I wouldn't be the least surprised that Oprah has such a friend in Gayle...that the 'hard stuff' goes in and out the listener's ears...right through, with no need for the listener to 'do' anything, unless requested to.
You know, if I were the young lady, I'd thank dad for offering a listening ear, and mention that the offer will be taken up when and if she feels the need. "You are a dear to offer...and I won't forget." And give him a little kiss on the cheek...or a hug...depending. I know that kiss will be remembered forever.
With the 'rules' being what they are, appropriately, you could give a client a phone number and a code phrase, "You talk, I'll listen."
(That worked beautifully for the ending of the movie, "Oh, God!")
Another great book is "How to Make a Man Fall in Love with You," by Tracy Cabot...which is far more than it seems...human communications. Bookstores have a hard time keeping it on the shelves.
And, just for fun, Carole Jackson's "Color Me Beautiful" is a fine thing to read, for all young people, and so true! (She wrote "Color for Men," too.) People have been known to get tears in their eyes, just from seeing a friend 'get their colors'.