I think this take is pretty much what enlightenment and post enlightenment philosophy attempted to do. Benthamites like Singer (and Bentham) do this quite literally: Identifying axioms (the utilitarian principle), deduce principles (utilitarian math) and strive (or argue for) consistency.

Utilitarians tended to end up with some pretty abhorrent extremities, by faithfully following their principles to some pretty weird places. If you want to be strict in your formalism, this means utilitarians can either accept stuff like Singer's infanticide or involuntary euthanasia arguments or they can reject the underlying principles.

More commonly (certainly the case for Rand, Trotsky and most other modernists), they go down a rabbit-hole of increasingly complicated fundamentals... The fundamentals (principles/axioms) need tweaking in order to support the complex conclusions. Most modernists wanted moral theories that also tied in neatly with epistemology, political philosophy and often economics. Being modernists, and committed to deductive reasoning, they wanted a tight basket of fundamentals to yield the whole thing. Things tend to get complicated.

I like your quote. Very apt. That period was, IMO, the last (and probably best) attempt at modernist epistemology. Trotski, Rand, Popper, Wittegenstein...

My 100-years later thoughts on that is that (a) Popper was right and (b) Popper (ironically) ended modernist, purely rational moral & political philosophy. Drwaing that hard line between scientific reasoning and the rest of philosophy was essential. But, it didn't change the fact that science (to reverse your quote) has nothing to say about philosophy.

Leftist philosophy made a U-turn following this period. Trotski was the last rationalist here, and subjective philosophies dominated self described leftist thought from then on. Rand's philosophy was de-rationalised and de-modernised before becoming the operative political philosophy she wanted. Popper's own collaborators (famously) Hayek were careful about Popper's lines and avoiding "pseudoscience," but their philosophies continued on to form the basis of post-Popperian pseudoscience. Pseudoscience in all but the most technical sense.

...Basically, postmodernism happened for a reason. Modernist thought kept hitting brick walls. IMO, these were the "phenomenon of a null result." Morals, politics and such just don't have a purely rational basis. No philosophy is both perfectly consistent and parsimonious with our moral senses.

At first pass, these conclusions lend to nihilism. But this is only true to the extent that we stay within the modernist frame. Morals either exist, and can be analysed in a strictly logical way, or they don't. It ignores the fact that we do, in actual reality, have morals.

This is what happens when you get too attached to a philosophical frame. The "works in practice, but not in theory" problem.

Analysing philosophies (especially enlightenment oriented ones) in this manner is kinda what most college courses do. It's getting tired though, and less popular. IMO, an honest take on this would be a "why every philosophy is wrong" course... but this gets trite fast too. If you set out to identify inconsistencies, it isn't hard. Once you get into defences, "why the philosophy isn't actually flawed..." it descends into something like high medieval philosophy. Boring and anachronistic stuff like "the problem of evil" but without god.

Problems in philosophy rarely get solved. We just move on eventually.