Sounds like we're confusing philosophy as investigation with philosophy as a way of living.
A large part of it is that, in the non-dual Shaiva Tantra tradition, there is a extensive experiential component to it. Someone who merely believes in a philosophy has not yet embodied the philosophy. Philosophical investigation is a crucial stage towards embodying the philosophy.
There is a Western philosophy that is also experiential. It's called alchemy. Here, the truths that were developed are considered the second of three stages of transformation. The last stage is plunging those principles back into the messiness of life, until the philosopher is a living embodiment of the philosophy. This last stage is called the "red stage", where the philosophy is tested in life and further distilled over and over again until it becomes alive once again.
So I'm not sure I buy your argument on that distinction. It seems more to me what you are saying is insufficiently applying and practicing the principles and fruits of philosophical investigation.
Disclaimer: My view of philosophy has heavy western bias because of the three classes I took (they were all about western philosophers).
As a side, I thought alchemy was a pseudo-chemistry aimed at finding elixir of life (immortality juice) and philosopher's stone (turning things into gold). I believe Newton was an alchemist himself and he harmed himself in his fruitless pursuit. I never knew there was a philosophy of alchemy. Please correct me if I'm wrong.
There are strains of alchemy that does concern itself with elixirs and transmutations of substances, but a discussion of that requires gnosis. (These strains, though, are interesting in that they are wildly proliferated in both Eastern and Western cultures).
Math is not as fundamental as people like to think it is :-D
As for non-dual Shiava Tantra, I can see why you would think it is a set of principles derived from religion. The View derives from empirical methods, reproducible, if not objective. (There is no such thing as a privileged objective view in a non-dual philosophy). Those truths form the a priori in which the rest of the View descends from. It happens to appear to be religious, but it is not religious that most people think it is. Christopher Wallis's book, Tantra Illuminated has several chapters on just the philosophical view, including how Indian philosophers have different ideas on what constitutes validity and proof. There is a section there where Wallis speaks about the non-dual Tantric View in terms of Western Philosophy.
I get that your view of philosophy has a heavy Western bias. It's for that reason that the authors of the original article is trying to champion greater diversity. Check out Wallis's book sometime, maybe just the chapters on the View.
I haven't thought through much of it, but it's interesting to me. Was this taught in the 3 semesters on philosophy? (What I'm really asking, was this something considered important enough to teach at an intro level to philosophy?)