It is difficult to answer this question just by looking at the works of Husserl himself, somewhat easier with Heidegger and Merleau-Ponty, and even with Sartre. But let's give it a shot. You can learn from anyone. Yet the main things we can learn from Husserl need to all be "under interpretation," and an interpretation that he and his orthodox followers would not accept. For example, yes we should "get back to the things themselves," and this means phenomena, i.e. the things and events in our experienced world as we experience them. Start with experience: this is a good philosophical rule, although I would like to allow this to include the experience of reading/thinking: experience does not have to be outside the library. Start from experience: this is the same call that is made by the American pragmatists, and that is a plus.
Further, we can agree that philosophy involves description of experience, and even, shocking for some, that the aim of philosophical inquiry is intuition of essences. However, unlike Husserl, the essences intuited are not eternal and unchanging. (I have much to say about such essences in other writings, and will only outline my position here). Essences are not universals in the traditional sense: indeed they are individuals of a sort. They even have local color, i.e. when one intuits an essence, say the essence of art, there is an experience involved that is not purely mental, that has a sensuous aspect. Essences are patterns in the field of experience, very specific patterns notable for their generative power and their, at least at some point in their careers, infectiousness.
It follows that, contra Husserl, and indeed, most other philosophers, with the possible exception of Heidegger, I hold that the difference between philosophy and poetry is not very great. For example it is not true that philosophy just deals with universals and poetry just with particulars. Rather, the essences that are perceived and described both in philosophy and in poetry have two aspects: a particular aspect and a universal aspect. Both philosophy and poetry attempt to achieve understanding through metaphors, except that in philosophy the metaphors are not always seen as such, and do not operate quite the way metaphors do in poetry. (Often the metaphors can be identified as key concepts in the definition, concepts which are best not understood in a dictionary definition way.) Intuition of essences is also not purely subjective, but neither is it purely objective: the subjective/objective distinction collapses here, both in philosophy and in poetry.
Essences are real as intentional objects of phenomenological or philosophical (the same thing) inquiry, but are constituted as they are intuited. That is, the discovery of them is also a making. The distinction between concept and reality to which the concept refers dissolves in the intuition and description of essences. Thus, doing something, for example Duchamp buying and displaying a urinal, can be a revelation and a constitution of essence even without language involved (also in Duchamp's case language is involved....that is almost inescapable.) Essences are patterns in experienced that are to some extent actualized by their definition, and the elaboration of this definition is further actualization.
Again, I agree with Husserl that a kind of bracketing is needed for phenomenological insight, but the result is more like what Kant saw as disinterested aesthetic experience, the essences being more like what Kant saw as aesthetic ideas. Both Kant and Husserl said that we should not think about the existence of the thing, except Kant talks about this in relation to aesthetic experience, i.e. experience of beauty, and Husserl in relation to phenomenological description. The point here is that they are really talking about the same thing. Phenomenological bracketing is aesthetic in Kant's sense.
The intuition of essences does not give us, contra Husserl, anything unconditionally or absolutely true or necessary (it gives us, in Kant's language, aesthetic ideas, not ideas of reason), but it does give us something which is as if such. It is important that this experience of "as if absolute" or "as if necessary" is not just a mistake. It is what gives meaning to existence.
Bracketing needs to be seen as a number of different kinds of operations, all of which can be useful, none of which absolutely necessary. The brackets can be around various things in order to achieve various results. Different kinds of bracketing sets the brackets at different places. For example, you can look at the world as if it were all a dream, and this can be useful in trying to achieve intuition of essences, since a pattern may emerge that would not if one always operated under practical assumptions about distinction between illusion and reality. You can bracket out the non-formal properties and only bracket in the formal ones, or you can bracket out formal ones and only bracket in the non-formal ones that have to do, for example, with social context. Toggling between the two, or even cycling between these perspectives and others, can usually bring better, more realistic results than sticking just with one.
Husserl's claim that philosophy advances no theories is only true on one level, i.e. that true philosophy is not going to result in something eternal and unchanging. However, it is not true on another level: for the result of any description of any intuited essence is a theory. That's what philosophical theories really are. (I make no claims here about scientific or mathematical theories.) As Morris Weitz taught us (his great largely unrecognized philosophical discovery), you have a theory of e.g. art if you have an honorific definition of art, but an honorific definition of art is not a theory of art in the sense of a definition in terms of unchanging necessary and sufficient conditions, and all attempts to define art in terms of that are really just theories of art in the honorific sense. I would only add that such honorific definitions if powerful or sound are based on an intuition of the essence of art and a description of that intuition. The openness of art as an open concept requires definition and re-definition...the very creative process of art requires it. This goes across the board for all philosophical concepts and all of the essences that philosophy attempts to describe.
Husserl also worried, according to Robert Solomon, whose essay "What is Phenomenology?" (Phenomenology and Existentialism ed. Robert C. Solomon, Littlefield, 1980, I am mainly thinking about here) that theories "always assert more than their data" (4). and this has no place in philosophy. It is true that honorific definitions always assert something more than the data by itself permits, for honorific definitions prescribe as well as describe: they look to the future and not just to the present. But then an intuition of an essence has little to do with data anyway. Data can be collected, but the gestalt that is the essence intuited goes beyond the collected data. It very superficial to think that one can just describe data and get anything that can be called knowledge.
Essences violate the distinction between the ideal and the real: they are real in a sense and ideal in a sense: they are quasi-ideal and quasi-real. But the idea of trying to be "theory-free" in the way Husserl wanted has the advantage of not allowing ones previously learned or currently dominant theories, for example being a Marxist or an Analytic Philosopher, to prejudice the results of eidetic inquiry. Solomon asserts that "presupposition-less" means that "any concept and any proposition can be reassessed at any point" but then that is just was philosophy is anyway and does not distinguish any one philosophical school from any other.
Husserl attempts to find a middle path between dogmatic axiomatic systems of philosophy and relativist ones like those of Dilthey and Marx which claim that there is no philosophy outside of socio-political context. This is wise but only as strategy not as the final story: in the end there is no philosophy outside of context, and yet bracketing allows for the kind of close observation, observation under conditions, that, in turn, allows for successful intuition of essences, intuition that can be challenged, revised, and enhanced when the brackets are moved. Relativism is true overall, but relativism is well set aside in the game of philosophy: much credit needs to be given to the moment of intuition, the moment of creative philosophical thought (this kind of thought by the way is not limited by any means to philosophy as a professional discipline or even to philosophy as a practice going back in method to such heroes as Socrates, Plato, Aristotle and Confucius.)
We can have our "demand for absolute truth" that Husserl wants, but only in the sense that we are talking about an experience of absoluteness or necessity that comes with intuition of essences, essences that are only as if absolute, necessary, eternal, etc. Of course we then give up Husserl's naive assumption that we can have truths that will be taken as true in any intellectual environment. The dream is not a waste however. In trying to achieve the universal through our honorific definitions of the subjects of philosophical inquiry we bridge over boundaries and fuse horizons.
I wholly support Husserl's attempt to avoid "philosophical theories in disguise." Philosophy to be deep must be skeptical and radical. Solomon gives an enlightening example. He writes that for Husserl "a description of consciousness is constantly endangered by the many metaphors and traditional philosophical theories that present an image of the mind as a mysterious container or stream...A philosophy that begins by taking these metaphors and theories seriously has, according to Husserl, based itself on presuppositions instead of pure description." (4-5) Rewriting this the way I want it: a description of consciousness which involves insight into essences inevitably involves, when successful, an intuition of essence, which inevitably involves a metaphor. The key is to recognizes that one metaphor is never enough and will never do forever. It is also to remember that no metaphor is ever fully exhausted. The best thing for philosophy to do is to take these metaphors as seriously as possible since they are the products of phenomenology, but not to take the tired, no-longer working metaphors as seriously as one once did. Learning how to see something serious as un-serious is the bracketing move called satire or parody. At the same time, one has to take one's powerful new metaphors seriously, at least for a time. That is what phenomenological pure description is, contra Husserl, all about.