I'm not in quite the right head space at the moment, but will try for a coherent explanation.
In the Chinese nuclear family, these days, the actual words "please" and "thank you" are rarely used by either parent or child. They're all implied in tone, context, and full body language. Among the extended family, it's almost the same; the slight difference depends on how close you are with your family members. Generally, the older generation, do not use "please" or "thank you" at all in family situations.
The philosophy (that holds true even now) is that filial duty mandates that your elders never have to say "please" and "thank you" to the younger generation, because it's the younger generation's duty to look after the older generation, without question (or at least, without quibbling or expecting something so "mundane"). The reason is that the older generation has sacrificed much / all to bring up the younger generation. IIRC, it stems from Confucius.
Even between close friends, where there relationship has developed to such that you're close enough to be considered adopted siblings or part of the extended family on either side, "please" and "thank you" are rarely heard.
But between people who do not have the type of relationship above, "please" and "thank you" are frequently used.
In the case of the video, barring the shock and relief, the girl called out something akin to "Dad!" The tone was terror and wanting her dad there. His response was "Don't be scared; you're okay now!" Wanting her father there -- in the Chinese context -- is saying "thank you": kid was terrified and found comfort in the security and gratefulness in knowing her dad was there.
