Algebraic aspects of the Goldbach conjecture

You might be interested in this article on Goldbach over function fields. The approach is rather geometric/algebraic, so it does pass your "steers away from hard analysis" test.

Here's a paper that might be considered a step in that direction. The authors construct an explicit family of polynomials $(F_N)_{N \in \mathbb{N}}$ such that the $N$th cyclotomic polynomial divides $F_N$ iff $N$ is not the sum of two primes.

http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P235.pdf