There is nothing wrong with the argument. The mathematics are quite simple and the conclusion is sound - scale cancels out.
Let's consider the essence of the question; How does the scale size of an animal affect the absolute height it can jump?
Let's assume an on-the-spot spring jump so we exclude a run-up. Now consider an arbitrary animal (let's call it a ballerina). It crouches, extends itself and leaps in the air. The difference in height of, say, its head between crouch position and when it leaves the ground is its leg extension ($s$). The difference in height between leaving the ground and top of the jump is the jump height ($h$).
As the ballerina jumps, its muscles provide an upwards acceleration of $a$ over the extension length, $s$ of its legs. 771
This leads to an initial upwards velocity, $v$ that is given by: $$ v^2 = 2as $$
As it flies upwards, it is decelerated by gravity until it stops at the top of its leap. The height reached is given by the same equation where the acceleration is that due to gravity, $g$: $$ v^2 = 2gh $$ hence $$ h = {v^2\over{2g}} $$
Since $v$ is the same in both equations, we can equate them, giving us: $$ h = {{as}\over{g}} $$
Now, $a = {F \over m}$, so: $$ h = {{Fs}\over{mg}} $$
We now have an equation for how high a ballerina can jump that depends on the force in its muscles, $F$, the length of its leg extension, $s$, its mass, $m$, and $g$.
What happens if we scale up the ballerina by a factor $x$? Leg extension simply goes up by $x$. Mass, which depends on its volume, goes up by $x^3$. Interestingly, muscle strength goes up by $x^2$ because it is the cross-sectional area of a muscle that gives it its strength (not its volume!). Plugging in the scale factors:
$$\begin{equation}\begin{aligned} h &= {{x^2 F \cdot xs}\over{x^3m \cdot g}} \ &= {{{x^3}\over{x^3}} \cdot {{Fs}\over{mg}}} \ &= {{Fs}\over{mg}} \end{aligned}\end{equation}$$
So we get the same height.
The conclusion is that the scale size of a creature is not a factor in calculating how high it can jump. To put it another way, the graph of jump-height versus scale size is a horizontal line.
Incidentally, elephants can actually jump - there is a circus stunt in which they stand on their two hind legs and hop slightly. They go up 10-20 cm, which is about the same as a flea. 771