How [do philosophers] strongly support or refute the view that: mathematics is a bag of tricks for real-world problem solving; undecidable statements are an irrelevant and harmless side-effect of an obvious consistency of axioms (capable of producing essentially only trivial results, however deep & wonderful they may appear); and mathematical platonism of eternal truths a psychological fallacy?
This is not a single logically coherent view, its a collection of views that have little to do with one another, other than being condescending toward mathematics as it is practiced. Some of them are contentious, some have followings and others appear to matter to a few individuals, some seem obvious to me, some are clear to no one. None of them are related.
The purpose of mathematics may in fact be to solve real-world problems, but that does not explain very much of how it came into being. Whole disciplines have been dominated by problems and practices with little or no practical value, like doing geometry with an irrationally restrictive set of tools, factoring all fifth-degree polynomials algorithmically, or determining whether there is a pattern to the structure of the transfinite sequence of ordinals. In fact, it is surprising to a certain swath of philosophers and philosophically inclined scientists (e.g. Eugene Winger) that the kind of math that arises naturally from such bizarre investigations is, in fact, useful. Some of them (e.g. Tegmark) find it so unlikely that they overreact and imagine that math must be what the world is made out of.
The 'obvious' consistency of mathematics has been in question since Frege failed to conquer Russel's paradox. Ways to stay clear of the inconsistent parts have been proposed by modern formalism, intuitionism and other schools philosophically, by interesting logical tricks like fictionalism in the middle ground, and by internal mathematical models like Woodin's attempt to describe the network of all possible models of set theory in a single metamathematical structure. Whether all this is pointless or not, people are going to do it. Potential inconsistency, even when just hinted at, is a human obsession. So why go out of your way to declare it pointless?
Whether mathematics has real logical content is a good question, but not related to its purpose, or to its consistency. Wittgenstein, notably, thought all of mathematics was in fact made up of tautologies. This is sometimes called mathematical trivialism. I agree with the reasoning. But very few people pay attention to it because it has nothing relevant to say. Tautologies can easily be so complicated that any given individual is quite unlikely to come up with them, and most of mathematics is at least that complex. So whether this is content or not, it historically enables science, and entertains mathematicians.
How to address the fact of the obvious impossiblity, but apparent necessity of Platonism is another issue altogether, and again, independent of all three of the prior ones. Most mathematicians still have to work in a way that presumes Platonism. But then they carefully formalize any results, because we know certain corners of logic fail. But we have no better reason to trust the formalism than the original framing -- they are both just stuff we made up. As an alternative, neo-Intuitionists have proposed the idea that mathematics is really a branch of psychology, investigating which of our most basic natural intuitions do and do not fit together passably well, including the specific delusion of Platonism, why we are led there, and what the alternatives would be. I hold this position. It makes sense that if we can work out a good basis for constructive mathematics, it will provide a single understanding that replaces two incomplete approaches. But overall, very few people care.
(There are a lot of random references in there, and I am not making them links, the SEP is searchable, so is this exchange, and Google works.)
The best way to do that would be to do what philosophers of the mind do, and reject historical forms of realism like objective reality, and Cartesian duality. Gilbert Ryle in his The Concept of Mind attacks Cartesian rationalism, and purports to show that the mind-body duality is a category mistake. In philosophy, the position that defends that psychology is source of knowledge is called psychologism. Robert Audi in his Epistemology, in fact, accepts that there are five sources of knowledge: perception, memory, consciousness, reason, and testimony. A single class in Psychology 101 would manifest the obviousness that these are the objects of empirical study.
The position in the analytical philosophical tradition is that scientific statements about language (pscholinguistics) and mind (cognitive science) should be admitted into philosophical discourse, and thus elevate the importance of empiricism in philosophical debate.
There is a long standing epistemic debate among various flavors of rationalism and empiricism regarding the certainty of knowledge, and this philosophical clash is highlighted with the Agrippan trilemma which strikes at the heart of three types of rational argumentation, the axiomatic or foundational, the circular, or those of infinite regress. Ultimately, psycholinguists and philosophers who align themselves with empiricism (Gilbert Ryle, Jaegwon Kim, Ray Jackendoff, Steven Pinker, Daniel Dennett, George Lakoff, and Mark Johnson, eg.) have leveled a series of philosophical attacks on objective realism in one way or another of various strengths. Dennett for instance is a physicalist who argues there is only the physical and that consciousness is an illusion. George Lakoff and Mark Johnson advocate a philosophy they call embodied realism which rejects objective realism entirely. If you want a detailed roadmap because you accept cognitive science as a path to truth, consider reading Philosophy in the Flesh by Johnson and Lakoff who articulate an ontology that accepts both supervenience and rejects Cartesian duality.