How I'd feel about my children being considered "our" (community's) children isn't relevant to the discussion of properly interpreting what this person was actually saying.
Back on topic, recall that the "these are our children" statement was her stating a notion that doesn't exist. But, to follow your question (and frame it properly), she does talk about recognizing that "kids belong to whole communities". As you say--she said what she said, not what you read into it. So, how do you know what she mean't by that statement? It's open to interpretation, based on contextual evidence. Which evidence shall we use...?
Clearly, this is the evidence you're using, which I'll grant you. I've never seen the show, nor am I familiar with this particular person's "agenda". I'm using the actual words in the video, where she talks about investing in education at the beginning of the spot, and making proper investments at the end of the spot. Without the same source from which you're basing your interpretation, you and I obviously will not come to the same conclusions. If you're going to debate her meaning based on external information that you have not brought to the fore, then clearly we're going to differ. Absent that--based solely on the 30-second video--I hold to my interpretation of her video, because it seems to stand alone.