How can you convince me that conservatives care?

Well you could start by answering my first two questions:

Since you're so worried about being misrepresented, what is your actual argument?

What kind of systemic national programs for children's welfare are you in favor of?

Note that that second question doesn't say "government program" (although I'd argue that that's the most efficient way to do it).

If you actually care about making sure that America's children are healthy and fed and educated and able to live effective lives, lay out the options that will actually work in the real world, public or private.

You will notice that the overwhelming majority of successful nations have affordable national healthcare and free public schooling, and don't just rely on private donations to make sure poor people, especially children, get to eat.

If you're going to argue that conservatives actually care, you have to provide conservative plans for addressing the issues at scale, and explain why they haven't been adopted or haven't been effective yet, despite the billions of dollars Americans give to charity.

Otherwise, you end up justifying a situation that actually leaves thousands of American children unfed, unhoused, and poorly educated, because people think that it's more important to maintain ideological purity than it is to actually use programs that have been effective in country after country.