Why is easier to measure directly the rate of increase of the volume than the rate of increase of the radius?

If $V = kr^3$ (assuming some homologous shape change then $k$ is a constant), then $$ frac{dV}{dr} = 3kr^2 = 3frac{V}{r}$$ $$ frac{dV}{V} = 3 frac{dr}{r}$$

i.e. the fractional change in the volume is 3 times the fractional change in the radius. If you have fixed fractional measurement precision - i.e. if your measurement error in radius is 1%, that leads to a 3% measurement error in volume. Or, you can say that the relative rate of change of volume is three time the relative rate of radius change.

You could just put the balloon on a weighing machine. The density of water is constant. You could measure $frac{dV}{dt}$ by $frac{dV(t)}{dt}=frac{1}{rho} frac{dm(t)}{dt}$, where $rho$ is the density of water and $m(t)$ is the mass of the water at time $t$ measured by the weighing machine. This relation can be obtained by differentiating $m(t)=rho V(t)$.

The above seems significantly more convenient than setting up a ruler and precisely aligning it with the center and end points of the balloon.