If we increase N type doping in a pure Silicon then according to the equation $$P_n=\frac{n_i^2}{N_D}\text.$$
Then mathematically, hole concentration is less than \$n_i\$, if \$N_D>n_i\$.
The holes (and free electrons) are constantly in the process of generation and re-combination. For intrinsic Silicon, the holes and electrons are equal in number since the (free) electrons are thermally excited from the valence band into the conduction band; i.e. for each electron in conduction band there is a hole generated in the valence band.
The electrons at \$t=0\$ are not the same exact electrons at \$t=\tau\$. This is because the initially generated electrons randomly recombine with holes and new electrons get generated/excited randomly. At equilibrium the thermal electron (or hole) generation rate balances the recombination rate.
In doped silicon, the excess conduction band electrons from the impurity shift the rate of re-combination. Each hole now has a larger probability of encountering a free electron to recombine. The generation rate is still the same since it is decided by the material and temperature (which are fixed for this discussion) (As you rightly pointed out). But the recombination rate depends on the probability of a hole encountering an electron. This probability increases due to increase in available electrons (from doping). So concentration of holes are expected to be lower than in intrinsic Silicon due to increased recombination rate.
Converse phenomenon happens for P type Silicon also.
Then mathematically, hole concentration is less than ni, if ND>ni.
The derivation of this equation already supposes \$ND >> n_i\$. page 4
To explain this phenomenon, we don't need Energy band theory. We don't even need electrons, holes or even Silicon. The phenomenon is purely driven by probability. To illustrate, we can take a similar phenomenon where Silicon is replaced by water, electrons and holes are replaced by H+ and OH- ions. Intrinsic Silicon is replaced by neutral water. Addition of base or acid to neutral water replaces doping. You can also draw parallels to pH of a chemical where addition of base (supplying OH- ions) reduces the concentration of H+ ions.
concentrations. First paragraph of the link
Adding an acid to water increases the H3O+ ion concentration and decreases the OH- ion concentration. Adding a base does the opposite. Regardless of what is added to water, however, the product of the concentrations of these ions at equilibrium is always 1.0 x 10-14 at 25oC.
