In a broad brush, at the heart of the difference between your comparison between the mathematical method and the philosophical method (whatever they may be) is embedded an analogy about language use. I would respond that it is a false analogy. Mathematical computation is fundamentally easier to understand and verify than natural language argumentation.
The primary difference between mathematical texts and philosophical texts is that mathematical texts approach theory building in a much different manner than philosophical texts on account that math is generally a deductive process built on an artificial language which uses a truth-conditional semantics whereas philosophy is built on defeasible reasoning including abduction using natural language using theories of semantics that look to be comprehensive by using conditions including but beyond truth conditions.
So, when Newton uses math to prove a theorem, and that theorem has been vetted for centuries the meaning of that theorem is rather indisputable. Its meaning is as consensual and clear as a meaning can get since mathematics uses a clear set of symbols and axioms when creating well-formed formulas (WFFs). In fact, WFFs are so consistent and unambiguous, that their usage can be formalized and automated such as those used in formal languages and automata, the latter being a class of mathematical objects which includes the Turing machine, the essence of a computer with a von-Neumann architecture.
Natural language and the defeasible reasoning in philosophy simply can't be reduced to the same formal system. Since this is the case, meaning is much more difficult to establish, agree upon, and verify. Whereas in the Netwon example, mathematicians the world over and can look at the calculations and agree, the same cannot be said from a passage from Wittgenstein translated to English and partially cited in the context of an argument. The relevant concepts in linguistics are paraphrase and metaphrase. In fact, the very nature of what constitutes the equivalency of propositions is quite a philosophical problem in and of itself and is naturally a function of your philosophy of language. One prominent term used in conjunction with the study of synonymy is Noam Chomsky's deep structure.
So in summary, the importance attached to the original propositions of the original texts is held generally by philosophers because of the ambiguity that inheres in natural language. Ultimately the nuances of language, whether those of implication, implicature, connotation, and denotation, are even complicated by figurative language such as metaphor. This is precisely why mathematicians insist on using formal symbols stripped of ambiguity, and undermines the idea that one can do the formal computation of mathematics (we are, of course, neglecting mathematical philosophy) and informal logic used in philosophical argumentation in the same way. This dream that Gottfried Wilhelm Leibniz held, the construction of a universal framework, and the ambition of eliminating metaphysics from science, an ambition of the logical positivists, have both been widely regarded as impossibilities.
Given my time constraint and capabilities, wouldn’t using secondary sources be a (comparatively) better idea?
In response to the more normative question of should you rely on secondary sources in philosophy, these are empirical facts:
- Often given a lack of extant work, you have no choice and must!
- The Western philosophical tradition is more than 2,000 years old and bigger than any one person.
- Professional philosophers themselves rely heavily on anthologies and textbooks which cite important selections from important works. Then, reading encyclopedia entries such as those at SEP is generally the next best step to understanding a topic and who has contributed to it.
- Ultimately, philosophy is useful to different people in different ways.
Of course, it's preferable generally not use primary sources if one is more interested in a comprehensible overview.
This matter is also an epistemic question, and is related to the notion of testimony.
Taking together these facts, it's hard to escape the conclusion that even heavy use of secondary sources isn't actually the NORM in philosophy broadly. As you note, there's simply too much going on to stay well rounded not to. In fact, as a non-expert in any field, the heavy reliance on secondary and tertiary literature over primary literature is the norm.
In my recent response to a question about Epictetus here on Philosophy SE, for instance, "primary" sources on Epictetus such are actually second-hand. No extant work exists from the thinker himself. To not use secondary sources would be irresponsible. Of course, a professional philosopher in this area will rely heavily on very specific primary, historical and philosohical sources. Other journals, contemporaries, archaeology, etc.
I purchased this philosophical encyclopedia sometime ago. Were I not saddled by adult responsibility and just read it, it would still take me an entire year to read all the volumes and make some sense of them if that's the only thing I did. Alas, I can't. And this set is essentially an attempt at a conspectus of literature.
Often, the best route to becoming somewhat of an expert on an area in philosophy is to read philosophers who select, interpret, and collate ideas around a topic professionally. Robert Audi, for instance, would be in a position to comment on epistemic matters far better than even most professional philosophers, so when you want to adequately paraphrase a concept, use the dictionary he edited which contains contributions from over 400 distinguished thinkers. If you get into a clash with a friend who has read the original work and Robert Audi who under the guidance of some experts in a field, it would seem to me that the testimony of the latter is superior from a heuristic standpoint. That goes double if your friend has an inadequate understanding of the principles of logic and language whatever her pedigree. Teachers and pundits are legitimate sources of truth in and of themselves. Audi's epistemological theory has an entire chapter that reasons why and how testimony (read secondary sources) should be used.
Lastly, and no one is an expert on your goals quite like you, if you're looking for a reasonable representation of what constitutes philosophy somewhere in between novice and expert, then why would you spend time reading most primary sources. It would be a good idea to read some of the important ones to get a taste (I personally think Uses of Argument by Toulmin should absolutely read by everyone interested in philosophy because of its views on informal logic and ordinary language). Perhaps for you given a background in finance, Wealth of Nations by Smith.
Do you absolutely have to read sources? To be a good professional philosopher, absolutely. To be a good dabbler, some is good. If you're just trying to understand the vocabulary for discussion at a party? Seems overkill. So here, like in all matters communication, context matters.