I understand your hypothesis and it may have an impact (or not) at the local level but according to gobbler the number of acres (on private) burned is very small. The biggest problem is using surrogate data (harvest) to predict populations. It simply may or may not be the case. Maybe longitudinally you could use it to support change in management (like 10 or 20 year trends), but otherwise it is just a snapshot of harvest and not necessarily population.
I
I get the point of birds returning to an almost like population sink kind of situation. And at least some literature seems to support that. Summary of the source:Fire and Turkeys in SE US

"Fire frequency: Fire frequency strongly affects plant community composition, and a fire-return interval of 1-3 years has been found to maintain the herbaceous
ground-flora desirable in pine-grassland communities. Turkeys, however, use a variety of habitat types throughout the year, necessitating a range of vegetative
conditions. For instance, during nesting stages, and during the first ~2 weeks after poults hatch, dense vegetation (typical in areas burned 2 years prior) is
desired for protection from predators. But, once poults begin roosting in trees, areas not burned in 3 or mores years are likely to be utilized. Similarly, during
the non-breeding season, turkeys more often use areas not typically burned, such as hardwood forests in bottomlands, drainages, and swamps. As such, the
variable habitat conditions required by turkeys throughout their annual cycle require sites with fire histories ranging from frequent to not-frequent fire."

"The authors note that valid
concerns exist regarding growing season burns which coincide with peak
nesting,[b] though studies show that females preferentially select sites that
were burned within the past two years, and as such do not typically choose
to nest in sites scheduled to burn.[/b] Also ameliorating potential effects of
exposure to fire, females exhibit a high propensity for re-nesting if a nest is
destroyed."

I haven't read that study, but this hints that return intervals of less than three years could be an issue. And the Authors agree-

"The authors suggest that growing season burns can be
appropriate when managing for turkeys, particularly
where managers are unable to meet objectives using
dormant season fire alone, though it is not
recommended to be employed at a return interval ≤ 2
years since these stands are preferred by females during
nesting and brooding periods. It appears that fire-return
intervals of ≥ 3 years are appropriate, particularly if
distributed in a mosaic throughout a fire-managed
landscape so to increase proximity of burned and
unburned areas. "

BUT...That being said, I stick with my original point and add one thing-I highly doubt fire is a major factor on county level populations outside of maybe a couple counties dominated by NF with large scale burns (if less than a 3 year return interval).