3 Responses to the question:
Why have those scientists who rejected or opposed philosophy, still succeeded?
Either (1) they are succeeding because they are correct in rejecting philosophy. (2) They are succeeding, but they are engaging in philosophy without knowing or acknowledging it. (3) They are succeeding because they are, per Kuhn, in the normal phase of science. If they were in the the revolutinoary phase or pre-paradigm phase they would not succeed unless they engaged with philosophical (metaphysical and epistemological) questions.
(1): Scientists who reject philosophy are correct, philosophy isn't relevant to science:
Those oppose philosophy as useless or no longer relevant to science are subscribing to a colloquial version of the philosophical view called Logical Positivism. Logical positivists held that the only meaningful statements are those based on logic and empirical facts (i.e. scientific statements). They were strongly opposed to metaphysics, which they considered to be literal nonsense since metaphysical statements could not be verified experimentally. And although they didn't reject philosophy all together (after all, they themselves were philosophers), they saw it merely as a set of linguistic and logic tools that served mainly a method of clarifying scientific statements and questions. Although Weinberg, Krauss, and co. don't agree on the details with the Logical Positivists, they are in the overall same epistemic ball park with them, in the sense that they view empirical observation as the only source of truth.
From this perspective, the scientists who reject philosophy are indeed correct. Philosophy is irrelevant (specifically metaphysics and epistemology) to their field, and that is why they are succeeding in their chosen filed.
(2): Scientists who reject philosophy are doing philosophy, they just don't know it (or won't admit it):
There are several criticisms of the Logical Positivist position, but in the context of your question, the relevant one is that claiming that the only meaningful facts are empirical ones and that non-testable metaphysical statements are meaningless is itself a metaphysical statement. So the scientists who reject metaphysics and epistemology as irrelevant to science are implicitly taking a philosophical position when doing so.
2.1. Dennett in his book "Darwin's Dangerous Idea" says: “There is no such thing as philosophy-free science; there is only science whose philosophical baggage is taken on board without examination”
Daniel Dennett gives a similar criticism of Lawrence Krauss's view (one of your rejecters of philosophy) in this debate between Massimo Pigliucci, Lawrence Krauss, and Daniel Dennett -- starting at 1:17:00 to about 1:24:00: Lawrence Krauss is engaging in metaphysics as much as any philosophically inclined scientist (Such as Roger Penrose or Lee Smolin). In his response to Dennett's criticism, he admits that he does engage in philosophical questions and philosophical thought experiments, and tries to give a more nuanced distinction between philosophy and science.
(3): They are succeeding only because they are engaging in what Thomas Kuhn calls the normal science phase as opposed to revolutionary science phase or pre-paradigm science phase. If they had to engage into some of the other phases of science, they would not be able to succeed without engaging with metaphysical and epistemic questions:
In his book "The Structure of Scientific Revolutions", Thomas Kuhn divides science in to 5 phases:
- The pre-paradigm phase: This is the primordial situation when a given scientific topic was based mostly on competing philosophical speculations. Think physics before Newton or psychology before the 20th century.
- The normal science phase: A specific paradigm (or theory) has been established and now scientists are engaged in confirming and elaborating on the consequences of that theory. Think Physics between the time of Newton and the late 19th century.
- The crisis phase: Experimental results start to seriously contradict the predictions of the theory established in the normal phase. Scientists will try to resolve these contradictions according to established theories. If they fail then science goes into the next phase. Think physics in the 1890-1910 period.
- The revolutionary phase: The previously established paradigm is being abandoned due to too many conflicting experiments and different paradigms and frameworks are competing to establish themselves as the new paradigm. Think of the period between 1910 and 1940 when Quantum Mechanics was being elaborated.
- Post-paradigm shift: Scientists agree on a new paradigm and start work on elaborating it and confirming it, thus returning to phase 2.
A similar classification is given by Imre Lakatos, who instead of paradigms, speaks of progressive research programs (when a theory is making definite progress) and degenerative research programs (when a theory is mainly defending itself from negative results - before finally having to be changed or abandoned).
Per 2.1, scientists can only ignore the philosophy underlying their scientific approach because they are in the normal science phase - or per Lakatos - in a progressive research program. That is why they are successful despite their rejection of philosophy - they are in a phase where the underlying philosophical assumptions are fixed and don't require any changes. If they were to find themselves in degenerative research program or in a crisis or revolutionary phase, they wouldn't be able to succeed without addressing metaphysical and epistemic questions.
Lawrence Krauss (again one your chief rejecters of philosophy) admits this in the debate video I linked to, although he doesn't use the terminology of Kuhn or Lakatos : When science is at an edge, then philosophical speculation is warranted. He says this about cosmology 40 or 50 years ago, and he also concedes to Dennett that studying consciousness and questions about the mind are much harder than physics because no paradigm has been established and much of the work is very philosophical in nature.