Given that Poisson is a special case of Negative Binomial which seems to just make error more likely in the case of overdispersion, without offering any real benefits, why would you fit a Poisson regression over a Negative Binomial? Is there some downside to fitting a Negative Binomial I'm just missing here? I guess it requires you to estimate an additional parameter, but when does that functionally matter? I'm sorry this question is somewhat open-ended, but I am puzzled why NB is not just a default recommendation.
For many practical applications the negative binomial distribution is more appropriate and is often a reasonable default choice. This is the case whenever we assume that risk varies across observational units (such as patients, hospitals, ...). The Poisson distribution may be appropriate e.g. when it is very clear that units are truly identical (e.g. identical atoms) and should have the same event rate.
It is rather easy to interpret as each unit having a Poisson distribution with the mean rate varying across units according to a Gamma distribution.
Very reasonable alternatives include a Poisson where the logarithm of the mean rate varies across units according to a normal distribution (i.e. a Poisson generalized mixed effects model with normally distributed random effects on the log-mean rate). This does approximate a negative binomial distribution reasonably well - a log-normal is pretty close to a gamma for suitable parameters, and let's be honest, we usually don't really know what distribution the event rate folllows across units.
The Poisson distribution has a very simple heuristic for its single parameter: the rate of occurrence of a rare event, with events happening independently.
Contrast that to the Wikipedia formulation of the negative binomial distribution:
In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of successes (denoted $r$) occurs.
Most scientists are well acquainted personally with situations involving many failures before a limited number of successes. Nevertheless, it can be hard to explain (for me, at least) what is going on with a certain set of observations that leads them to follow a negative binomial distribution. The rate in Poisson is much easier to interpret in physical terms, despite the sometimes counterintuitive appearance of a set of independent events.
So in the spirit of "all models are wrong but some are useful" one might prefer to start with Poisson and only move on to a negative binomial when it's clear that the Poisson is inadequate.