There's a few different questions (at least from the perspective of those who give natural law some role) that are tangled up in your question. And if few parse some of these, Holmes' views should be clearer (for full disclosure, I haven't read Holmes but have studied the natural/positive law distinction within philosophy).
How does something become a law for a given nation?
Positive law answer: judges or legislators make it a law for that nation.
Natural law answer: people use reason and look at nature and then make that into a law for their nation.
In a sense, the positive law answer is cleaner because it bypasses why, and the natural law answer to some extent agrees completely with that why-less answer: the laws of a nation are the laws a nation gives itself.
Question #2 What justifies laws?
Positive law answer: that someone with power made it into the law.
Natural law answer: that it matches what our use of reason and our insight into nature tells us should be the law.
Here, the positive law theorist can argue that the natural law theorist is also committed to his answer. Something only becomes a law when someone has power, and in a sense this is the justification of any law. E.g.,
Positive law justification: "why should I do what you're telling me?" "Because I will take away your TV time, and I can because I set the rules around here."
Natural law justification: "Why should I do what you're telling me?" "Because the concept of man is a rational animal who needs to develop his or her faculties of reason and the body in order to reach a state of true fulfillment. Oh, and if your reason doesn't make that sufficiently clear to you yet, then well, I will discipline you by taking away your TV time until you attain the state where you can manage your virtues and vices in accordance with reason which is a part of your nature not yet fully realized."
I think these two questions enable us to better address the claim of the legal positivists.
I'd suggest instead of "all law is positive law" that what we wind up with is:
if the justification for law is the power to enforce it, then all laws are positive laws.
This contains a condition, the positive lawyer accepts but the natural lawyer does not.
that something is the law means that it is a positive law (in the mode of how it became law)
which is trivial.
Question #3: So, why then do legal positivists not care that there are alternate answers to legal questions?
So there's natural law and rational constructivism and divine command theory, but these are not at all convincing for the legal positivists. The reason is that legal positivism is generally joined to a commitment that all we do when we work with language is shuffle terms.
Ergo, the attempts to show that there's any real sense of justification (i.e. a metaphysical reason why law ought to bear a certain character) is just meaningless jibber jabber masquerading as rationality. Similarly, any attempt to find a basis for law in nature is again just cultural biases and other things masquerading as something more than they are.
In other words, legal positivism is generally committed deeply to an anti-realist reductionism, and consequently it doesn't consider what others do to be what they purport to do but rather only purporting simpliciter.
(if the OP wants to clarify that he wants a specific explanation of the exact way Holmes argues this -- then this answer isn't fitting. I'm answering what I take OP to be asking -- which is why does Holmes not think there's any other kind of law?)
tl;dr - the degree to which you find Holmes et al.'s claim that all law is positive law convincing is basically linear with the degree to which you find the underlying claim that we're not connecting with reality with our "reasoning" convincing. The less convincing you find that claim, the less convincing positivism. The more convincing the claim, the more obvious positivism about law becomes.