The math is correct.  If p = the private key, P = the public key and G = the base point of the EC crypto system then P = p(G) where p(G) is the scalar multiplication of p selected from the finite field used to generate the elliptical group with the base point G from the group.

So for any two key pairs: P0/p0 and P1/p1

P0 = p0(G)
P1 = p1(G)
P0 + P1 = p0(G) + p1(G) = (p0 + p1)(G)

So you can take two public keys and apply the defined elliptical group addition and the resulting public key will have the corresponding private key which it the simple modulus addition of the two private keys.

Do you like ketchup/catsup (spelling looks funny) on you underwear?

This is not voodoo it is fairly simple finite field math and group theory.  Check the algebra above.  Do you see any mistakes?