You can do something rough using homology relations which assume homologous structures for stars governed by a similar polytropic relation between pressure and density. So this will work for stars like the Sun and more massive where we can assume energy is transported radiatively and the structure of the star is governed by a $n=3$ polytrope.
The derivations are still quite lengthy, I'm not going to reproduce them; you can find them in this set of lecture notes for example.
The basic result is that the central density scales like
$$\rho \propto M^{2(3-\nu)/(3+\nu)}\ ,$$
where $\nu$ is the power law index governing the sensitivity of the nuclear burning rate to temperature: $\nu \sim 4$ for pp burning and $\nu \sim 20$ for the CNO cycle.
Thus $\rho \propto M^{-2/7}$ for pp burning stars with mass less than about $1.5M_\odot$, but density falls more steeply as $\rho \propto M^{-34/23}$ in more massive CNO burning stars.
Short of going through the stellar structure equations (which the link above does), a handwaving physical argument would go as follows.
The virial theorem easily tells us that the interior temperature of a star $ T \propto M/R$.
If the energy generation rate is strongly dependent on temperature and steeper than the mass-luminosity relationship, which it certainly is for the CNO cycle, then the central temperature hardly changes with mass and so $R \propto M$.
From this we see that density scales as $M^{-2}$.
In practice, the central temperature does rise slightly with mass, so $R$ does not quite rise linearly with mass and so the density decrease is not quite as steep. For the pp cycle, $\nu \sim 4$ is only slightly steeper than the luminosity-mass relation, so central temperature rises significantly with mass and hence the central density only falls slowly with mass.