Proving the snake lemma without a diagram chase

You can always "diagram chase" in any abelian category, without invoking any embedding theorem, using arguments with subobjects, as in MacLane's book.

In any case, you can also construct the boundary map as follows:

We are given a map $b: B \to B'$. Let $B'' \hookrightarrow B$ denote the preimage in $B$ of $\ker c$. (If you want to desribe this in more categorical terms, it is the kernel of the composite $B \to C \to C'$.)

Then the map $B''\hookrightarrow B \rightarrow B'$ factors through the monomorphism $A' \hookrightarrow B'$ (using the fact that $A' =\ker(B' \to C')\, \, $). This then induces a map on quotients $ B''/A \to A'/\operatorname{im}A$, which is precisely the desired map $\ker c \to\operatorname{coker}a.$

Checking the various exactness claims is just a matter of using all the relevant universal properties of kernels, cokernels, quotients, etc.

There's a completely element-free proof in Kashiwara's "Categories and Sheaves" (Section 12.1).

The salamander lemma described by George Bergman and summarised by nlab & the secret blogging seminar help simplify proofs of the basic diagram chases, including the 3x3, four, snake and long exact diagrams.

However the Secret Blogging Seminar says:

If you don’t like diagram chases, it’s likely that you still won’t like them once you know the Salamander lemma. The salamanders chase the diagrams for you, but you still have to chase the salamanders. I think the salamander proofs are easier to explain (once you know the Salamander lemma), and it’s easier to see where you use the hypotheses. For example, it is totally clear that the argument for the 3x3 lemma can prove the 20x20 lemma as well.