How is philosophy of science still one of the most essential fields after it broke from classical philosophy and established its own discipline?
This is my opinion. But then, any answer you get on this subject will be someone's opinion.
Philosophy in my opinion, is the asking and seeking of answers to ANY question. Whereas science is a branch of philosophy which arbitrarily constrains its area of enquiry to ideas which 'predict'. It does this regardless of legitimate problems with 'abduction' or 'empiricism' as a means of acquiring knowledge. It does this regardless of legitimate questions as to the exact nature of existence. It does this because so far it has produced staggering results. So staggering in fact that some areas of science have "almost" exhausted all areas enquiry (Chemistry being one such field).
And in fact any perceived split between 'physics and metaphysics' is narrowing. The work of Heisenberg etc. has had the practical effect of making physicists less 'reductionist'. It has demonstrated areas where the scientific method ought to be able to work, but just cannot. Physics has begun asking some old 'metaphysical' questions.
I recently watched this video which features Brian Cox. Many areas of this conversation stray deeply into a kind of metaphysics that science would hitherto have not really considered.
As Dr. William Lane Criag pointed out, science cannot sustain itself without some governing principle. We all thought (with the logical positivists) that the "Verification Principle" could do the trick seventy-five years ago...
THE PRINCIPLE OF VERIFICATION
The principle of verification is supposed to furnish a criterion by which it can be determined whether or not a sentence is literally meaningful. A simple way to formulate it would be to say that a sentence had literal meaning if and only if the proposition it expressed was either analytic or empirically verifiable.
Language, Truth, and Logic by A. J. Ayer, p. 5
...but we can see now that the proposition that I reproduced above doesn't satisfy its own criterion, and thus fails. For reasons like this, philosophy of science is indispensable in providing the governing principles that extend, retract, or enforce, as needed, the bounds of what science is from day to day.
Even if people stop calling themselves "philosophers of science", somebody needs to make defensible statements in support of science when other smartypantses come around trying to undercut scientific findings, and those statements will have all the features that identify the discipline of philosophy to us today.
If what you're talking about is that set of principles that is sometimes called "philosophy of science", but is also sometimes called "Scientism," metaphysical naturalism, scientific materialism, ontological naturalism, philosophical naturalism, etc. then I don't have a specific argument for why this is indispensable.