*What are some interesting examples where evaluating an expression assuming its existence is much easier than proving existence?*

Simple examples are given by limits of several variables (because we can often find a path along which the limit is trivial). For example, assuming the existence of
$$\lim_{(x,y)\to(0,0)}\frac{xy^2+\sin(x)x^2}{x^2+y^2}\qquad\text{and}\qquad \lim_{(x,y,z)\to(0,0,0)} (x^2+y^2+z^2)^{x^2y^2z^2},$$
we obtain
$$\lim_{(x,y)\to(0,0)}\underbrace{\frac{xy^2+\sin(x)x^2}{x^2+y^2}}_{:=f(x,y)}=\lim_{y\to 0} f(0,y)=\lim_{y\to 0} 0=0$$
and
$$\lim_{(x,y,z)\to(0,0,0)} \underbrace{(x^2+y^2+z^2)^{x^2y^2z^2}}_{:=g(x,y,z)}=\lim_{x\to 0} g(x,0,0)=\lim_{x\to 0} 1=1.$$