Logically, if the only proof of existence we have is what we can see, hear, smell, touch, taste, and feel the movement of, why do many philosophies deal with issues outside the physical? Due to these scientific limitations of not being able to prove anything nonphysical (not limited to entities/ concepts surrounding spirituality and religion[s]) exists in any concrete way that affects our daily lives in ways that WE OURSELVES don't choose it to, why doesn't philosophy eliminate these concepts?
I believe that in ordinary language philosophy there is a means whereby some of the "non-physical" terms can be analysed from a logical positivist perspective (which is essentially the philosophical school who take the view you have outlined in your question). The argument against logical positivism is a weak one and was only really allowed past without proper scrutiny because fashions in philosophy changed and logical positivism presented a threat to popular philosophers. The philosopher Michael Friedman describes the natural movement of ideas in philosophy as "quite inadequate" to explain the violence with which Logical Positivism was rejected, and so I think it is still relevant to questions like this with some modification.
To answer the question you only need to use an ordinary language philosophy approach to do work on metaphysical concepts. If instead of asking "what is God" or "what is happiness" you ask "what do we mean when we use the word God/Happiness?" Then you have a question which is verifiable by reference to real world use of language. With regards to Atoms and bacteria as mentioned above, metaphysical ideas relating to these concepts would, indeed, have been useless until methods arose for verifying them. Popper's scientific methodology is based on falsifying the simplest theory first (i.e. some empirically verifiable concept, not a wild metaphysical speculation). Even though Popper was opposed to Logical Positivism, his main objection was against the methods used to test theories, not the proposition that metaphysical theories were meaningless unless testable.
As an aside, the statement that was used to undermine logical positivism was that "The criterion which we use to test the genuineness of apparent statements of fact is the criterion of verifiability." is in itself a non-verifiable statement. This is an invalid argument partly the same grounds (the meaning of all the words in ordinary language is verifiable) and partly because of the context in which the statement is made. What it means is that there is no utility in in using criteria other than verifiability to test genuineness. Under a Utilitarian approach, therefore, this statement can be falsified by demonstrating any utility of metaphysical discourse other than that which has been proven empirically.
So, to relate back to your question, in some schools of thought, philosophers have no cause to examine the sorts of non-physical concepts that you describe unless they are interested in the ways we use the words that describe them, or there is sufficient empirical evidence to merit developing a theory about their existence which scientists can then attempt to falsify.
Well, your question seems quite interesting. My personal point of view on it is that philosophy is the art of dealing with thinking, especially questions that has great impacts on how we perceive life, or how we act day-to-day.
So the answer would be: because although sciences has not clearly shown the existence of a superior intelligence (like a god), religion still continues to exist and to have adepts. Then I believe you can't discard a subject if it is in an actual context and concern, and moreover as, furthermore, some people do believe in it at the very precise moment we're talking.
Truly, there is reflexion that can be done on things you can't prove by calculus or common sustainable logic. A common example would be the Descartes' statement: Cogito, ergo sum (I think so I am). Descartes points out his only certitude is that he exists as a human-being doted of the faculty of reflexion. That points outs a certitude that cannot be affordably demonstrated by sciences' patterns.
How can philosophy deal with the nonphysical in light of scientific limitations?
Primarily, respect for obtaining knowledge (read: philosophy) contends with the non-physical by recognizing the distinction between the physical and the non-physical (see "the excluded middle"). Epistemic limits do not preclude speculation and supposition, nor do they render axiom and self-knowledge invalid, nor language meaningless.
Logically, if the only proof of existence we have is what we can see, hear, smell, touch, taste, and feel the movement of, [then] why do many philosophies deal with issues outside the physical?
Proof is merely a means of evaluating conjecture as truth. If by "philosophies" you mean "ways of looking at things" then I can only offer that if philosophy were a way of looking at things, then an oasis and a mirage would have epistemic and ontological equivalence. They do not.
Due to these scientific limitations of not being able to prove the spiritual exists, why doesn't philosophy eliminate these concepts?
What is meant by 'spiritual'? If, as etymology suggests, you mean "breath" and both the conscious as well as autonomous functions by which we breathe without suffocating while we sleep, then the purview for inquiries is biology. If you mean self, then there is psychology, neurobiology and the philosophy of mind to address your concerns regarding spirit. If you mean a transcendental or immortal soul, you have religion and whatnot. It is not the task of philosophy to eliminate concepts, instead the virtue of wisdom can help focus your investigations by understanding what you are investigating. Hope that helps!