Recently for a bonus homework assignment in my algebra class, I was asked to review the literature and write up a proof that $\pi$ is transcendental. Essentially every source I found ("The Transcendence of $\pi$" by Steve Mayer for example) presents the classic proof of Lindemann, which heavily relies on symmetric function theory and in particular the fundamental theorem of elementary symmetric functions.
Before this assignment, I did not know symmetric function theory, and by far the biggest difficulty in my solution and write up was understanding this theory and the argument used in the proof (which in my opinion, was not spelled out enough for a beginner to the theory to easily understand the argument in the sources I consulted).
Now, symmetric function theory was useful to learn, and I am aware it is very useful in the proof of the Lindemann-Weierstrass theorem, but it begs the question: Is there any proof (preferably understandable to approximately a beginning graduate student) that $\pi$ is transcendental, without using symmetric function theory, and if not, is there a theoretical explanation for why?
Edit: Crossposted to MO after 2.5 weeks https://mathoverflow.net/questions/466288/proof-pi-is-transcendental-without-symmetric-function-theory