Another option is the Wilf-Zeilberger (WZ) method via Zeilberger's algorithm.

Start with the "seed" function (your summand)
$$F(s,m):=\frac{(-1)^m(a+2m)\binom{s}m}{\binom{a+m+s}{s+1}s!(s+1)!}.$$
The algorithm generates a "companion" function
$$G(s,m):=\frac{(-1)^{m+1}\binom{s-1}{m-1}}{\binom{a+m+s-1}ss!^2}.$$
Then, we ask a computer algebra software to verify that $F(s,m)=G(s,m+1)-G(s,m)$: one easier way is to divide both sides by $F(s,m)$ because the task would reduce to testing equality between two rational (or polynomial) expressions.

Next, sum over the integers $m\in\{0,1,\dots,s\}$ to obtain: $\sum_{m=0}^sF(s,m)$ which is the LHS of your claim, while $\sum_{m=0}^s(G(s,m+1)-G(s,m))$ telescopes to $G(s,s+1)-G(s,0)$. A quick check shows $G(s,s+1)=0$ and that $-G(s,0)=0$ unless $s=0$, in which case $-G(s,0)=1$.

In summary, we arrive at $\sum_{m=0}^sF(s,m)=\delta_0(s)$.