The Weierstrass $\wp$ function satisfies the addition formula

$$\wp(z+Z)+\wp(z)+\wp(Z) = \left(\frac{\wp'(z)-\wp'(Z)}{\wp(z)-\wp(Z)}\right)^2.$$

Of course, this is just the $x$-coordinate of the sum of the points $(\wp'(z), \wp(z))$ and $(\wp'(Z), \wp(Z))$ on the Weierstrass elliptic curve $y^2=4x^3-g_2x-g_3$. If one has an *a priori* knowledge of this fact, the computation of the addition formula is absolutely trivial. However, it seems that, in the literature, there is no "illuminating" proof from first principles. Perhaps my standards for "illuminating" are too high, but in comparison with proofs of addition formulas for trigonometric functions, the addition formula for $\wp$ is painstaking to establish.

I would like to know how the formula above, or an equivalent formula (perhaps for the Weierstrass $\sigma$-functions?) may be deduced in the most direct possible manner. Of course, a proof such as one that involves the comparison of power series around the origin is very direct, but it requires a first-hand knowledge of the formula, so it's not quite what I'm looking for.

All the best, and thank you!