The claim we set out to prove is the Rothe-Hagen identity
$$\sum_{k=0}^n
\frac{x}{x+kz} {x+kz\choose k}
\frac{y}{y+(n-k)z} {y+(n-k)z\choose n-k}
= \frac{x+y}{x+y+nz} {x+y+nz\choose n}.$$
We prove it for $x,y,z$ positive integers and since the LHS and the RHS
are in fact polynomials in $x,y,z$ (the fractional terms cancel with the
corresponding binomial coefficients e.g. $\frac{x}{x+kz} {x+kz\choose k}
= \frac{x}{k!} (x+kz-1)^{\underline{k-1}}$ as long as $x+kz\ne 0$
(consult problem statement)) we then have it for arbitrary values (we
also get polynomials when $k=0$ or $k=n$.)
Consider the generating function $C(v)$ that satisfies the functional
equation again with $z$ a positive integer
$$C(v) = 1 + v C(v)^z.$$
We ask about again with $x$ a positive integer
$$[v^k] C(v)^x = \frac{1}{k} [v^{k-1}] x C(v)^{x-1} C'(v).$$
This is by the Cauchy Coefficient Formula
$$\frac{x}{k\times 2\pi i}
\int_{|v|=\epsilon} \frac{1}{v^k} C(v)^{x-1} C'(v) \; dv.$$
Now we put $C(v) = w$ and we have from the functional equation
$$v = \frac{w-1}{w^z}$$
which yields
$$\frac{x}{k\times 2\pi i}
\int_{|w-1|=\gamma}
\frac{w^{zk}}{(w-1)^k} w^{x-1} \; dw
\\ = \frac{x}{k\times 2\pi i}
\int_{|w-1|=\gamma}
\frac{1}{(w-1)^k}
\sum_{p=0}^{kz+x-1} {kz+x-1\choose p} (w-1)^p\; dw
\\ = \frac{x}{k} {kz+x-1\choose k-1}
= \frac{x}{x+kz} {x+kz\choose k}.$$
Note that this yields the correct value including for $k=0.$
Now starting from the left of the desired identity we find
$$\sum_{k=0}^n [v^k] C_z(v)^x [v^{n-k}] C_z(v)^y
= [v^n] C_z(v)^x C_z(v)^y = [v^n] C_z(v)^{x+y}.$$
This is the claim.
Remark. The above does not quite fit the OPs request for an
elementary proof. It can nonetheless be made to work with formal power
series only by replacing the complex integral with the appropriate form
of the Lagrange Inversion Formula. Consider it an incentive for activity
on the question.
Remark, Mar 14 2020. For the LIF computation we put $D(v) =
C(v)-1$ so that we get the functional equation
$$D(v) = v (D(v) + 1)^z.$$
Using the notation from Wikipedia on
LIF
we have $\phi(w) = (w+1)^z$ and $H(v) = (v+1)^x$ and obtain
$$\frac{1}{k} [w^{k-1}] (x (w+1)^{x-1} ((w+1)^z)^k)
= \frac{x}{k} [w^{k-1}] (1+w)^{kz+x-1}
= \frac{x}{k} {kz+x-1\choose k-1}.$$
This matches the first result.
