I find it useful to approach this question from the bottom up. You made a philosophical argument:
My argument is that there is no way to deny the sure success of the sciences (like Physics, Chemistry, Medicine, etc).
One of the things I love about philosophy is once you take a stand like this, there's always a line of questioning which can unsettle your position. I find it's useful to unsettle it first, and then use that uncertainty to explore the other interesting parts of the question. So that being said, I will not deny the sure success of the sciences.
Instead, I will question it.
I'm not going to say that the sciences haven't had sure success, I'm simply not going to immediately accept that the sciences have been a sure success. I'm going to ask you to convince me. How sure are we? You will almost certainly point to the wide acceptance of science. Science is everywhere in our society! To which I will reply that bacteria are everywhere as well. Does that mean bacteria are a success? Is it really good if something succeeds, simply because "succeeds" is the word we used?
Are we happy because of science?
Do we have meaning, because of science?
Are we better people because of science?
These are harder questions. There are a lot of very strong arguments suggesting that science and technology have actually made us less happy, and further from meaning. After all, that's the general pattern which drives people to become monks, and monasteries are alive and well today. They're "succeeding" too.
Now I, myself, tend to find the science arguments compelling. I'm choosing to be the person questioning science here, but I understand where you're coming from. What I'm really picking at is
I think the scientific method must be assumed as an axiom/premise in Philosophy
(Emphasis mine). Such wording is a very powerful claim when speaking in the philosophical world. And, generally speaking, powerful claims are very hard to justify in philosophy. And, in fact, you will find that the scientific method actually does not always appear in every philosophy. I once attended a great lecture on what Traditional Chinese Medicine is, and the speaker explained a difference in how their tradition is developed as compared to Western medicine:
- Western medicine tears the body apart into components, develops
hypotheses about these components, then build them up. At each step,
it develops testable hypotheses and tests them. From there, it finds things which may provide results, and tests those.
- TCM starts with the body as a whole, finds things that cause good results, then develops testable theories about why the results
There's clearly a difference in methodology. Many would claim that TCM does not follow the scientific method. But TCM is deeply steeped in Chinese philosophy, so a claim that philosophy must assume the scientific method as a premise invalidates 4000 years of Chinese philosophy, declaring it to "not be philosophy because it doesn't assume the scientific method." You can understand why such a claim is not popular.
Or you might claim that TCM has the scientific method of hiding in it. As it turns out, there is no one "The Scientific Method." We talk about it as though there is only one, but there isn't. It's actually a rather large class of approaches ranging from the very precise definitions that are used at the LHC when making observations of subatomic particles all the way out to a very broad "guess and check" mentality. You could choose to define the scientific method to be broad enough to admit Chinese philosophy, thus making it easier for others to accept your claim.
But there's a catch. The weaker you elect to define the scientific method, the harder it is to have absolute confidence that it must be a good thing and a successful thing. It's actually a really fun exploration, which digs at the heart of what it means to be "science."
This is the Philosophy of Science and the point where I can segue to answer your original question. The philosophy of science is considered to be a subdiscipline of empiricism. Empiricism is the study of that which we can "know" with our senses, i.e. empirical observation. There are other subdisciplines of empiricism as well besides science. Empiricism, itself, is a subdiscipline of epistemology, which is the study of what we can "know." Knowing is an incredibly complex and nuanced thing in philosophy because it has to actually address questions along the lines of "How do we know that we know something? How do we know that we know that we know something?" and so on and so forth.
Epistemology, itself, can be contrasted to ontology. Ontology is the study of what is real. Most of us assume that we know what "reality" is, but ontologists really question it. It turns out to be a tremendously interesting topic, but it's almost completely divorced from science.
Which is interesting. We often have the opinion that science is a tool which leads us to the truth of reality, but when you dig far enough down into philosophy, you start to see that that isn't quite what it does. In fact, some even argue that it does the opposite: instead of leading us to the truth, it tries to lead the truth to us.
But that's just one argument. The point being, that philosophy explores a class of questions and their answers which reach far broader than science does. And that's a good thing. Diversity is good. Science is great at what it does (that is to say, science is great at being scientific). Philosophy is great at what it does (that is to say.... whatever philosophy does. It's actually a rather funny exercise to try to pin down precisely what philosophy does). Both are valuable in this world, and neither should really replace the other.
Now I have pointed out how much separation there is between science and philosophy. But there is a very important connection between them: philosophers are people. Their ideas mingle with the ideas of the population around them. While there is no explicit connection between science and ontology, you will find that many ontological philosophers strive to develop their research to fit nicely with the ontological claims that science sometimes makes (when they technically can't make such claims, being empirical). Other philosophers will explicitly strive to make their work conflict with science's claims, just to explore that "what if the world isn't the way we think it is" sort of question. Their philosophy will naturally reject the claims that the scientific method must be assumed because the point of their work is to see what happens when you reject it.
So science absolutely influences all of philosophy, though often in a very indirect way. Likewise, philosophy forms the cornerstone of what science, itself, is. But many find it's digging and questioning to be a bit of a bore, so we often ignore this connection to philosophy.