The original situation we're discussing is straight-forward and there's no need to delve into bizarre hypotheticals and risk confusing things further.

yet arbitrarily* removing opportunities for what you may
have in the future is perfectly acceptable

You talk about removing opportunities from people and this is mistaken. In the child situation, nobody has removed opportunity from the child. You have again tried to twist things so as to describe the natural order as being one where all charity options available to help a poor person realise all opportunities are exhausted. You then use the word 'stealing' to describe failure to reach that point. It's flawed and you don't win points for being original in trying it on. [1]

I'm not talking about government, I'm talking about
the morality of poverty itself, specifically in
children who have no responsibility for their situation.

But you are, because you're talking about rights, and the idea of a right that cannot be realised is invalid. There are some philosophers who are of the opinion that there is such a thing as a right but where there is no moral obligation upon anything else to guarantee that right. For example - the right to work. In contrast, my opinion is that if philosophers want to masturbate then they should do so in the privacy of their own home and avoid staining their work with it. If there is a right, then it is to be enforced, and if something should not be defended then it is not a right.

Suppose you had access to a magic button, and
pressing it will -- at no cost to anyone -- give
every child a fair start, where their life's success
will solely be the product of the wisdom of their
choices. Is there a moral imperative to press that
button? Would it be wrong, in the way that theft is
wrong, to turn your back on it?

What you're generally asking is an interesting question, but it's not pertinent to this discussion for these reasons:

* You've used the word 'fair' and I've already explained that as a worse-than-useless word in a conversation about public policy.

* You've mentioned "at no cost to anyone" and that's not what we're talking about here and it is a crucial part of the discussion.

If yes, then what we end up with is a debate along
utilitarian lines: At what cost must we address the
wrongs done to those born into poverty?

Sure enough - after having claimed that you're not talking about the bridge into coercion through government - you come clean and admit that actually that's exactly what you're doing.

If no, well, that is where I throw the 'nasty names'
around.

I suspect that this conversation will end in value incompatibility which is a shame, because I suspect if it does then at least one of us is not being entirely honest with self - we seem to both be broadly utilitarian. I think nasty names (which I fall to sometimes as well) are a temptation in two scenarios:

* Value clash with the other person, you make a last ditch effort to shame them into converting to your values.

* Situation in which you feel yourself having cause to abandon something you hold dearly due to internal incompatibility (e.g. someone who is a both religious and a white supremacist is having trouble reconciling parts of the bible's teaching with the fulfilling feelings they associate with Friday night ceremonies), and you want to pretend that there's a value clash to allow for an easy exit out of the conversation and being able to escape the need to reconcile the incompatibility.

p.s. : A total side issue, but I don't see that
taking stuff off a dead person is "stealing" from
anyone. ;)

The property belongs to them. They leave instructions on what is to be done with it. It is not unowned property, nor is there any doubt about the wishes of the owner in the presence of a will. Therefore doing other with it is stealing. In the absence of a will - well - that's up for debate, but it's not particularly significant in the grand scheme of this discussion.

[1] See Hayek's _The Road To Serfdom_. Chapter 2 is particularly useful for exploring this "characteristic confusion of freedom with power".