There are well over 2 million within 100 miles of Nashville. The Nashville metro alone has over 1.6 million and it abuts the Clarksville/Ft. Campbell, KY metro to the north (300,000), the Cookeville micropolitan to the immediate east and the Columbia, TN micropolitan to the south. The Huntsville metro is also within 100 miles of Nashville as well. Huntsville is essentially equidistant between Nashville and Birmingham.
The important thing for pro sports are the media market and corporate sponsorship ability. In regards to the media market, Birmingham actually has a larger DMA market than some cities with pro sports like OKC, New Orleans and Memphis. What hurts Birmingham more than anything is its proximity to Atlanta and Nashville who have multiple sports.
I actually thought of being more specific with my population analysis, I knew it would come back to bite me. Most of the population in the Huntsville area sits closer to Birmingham than Nashville. Birmingham also has the added benefit of having all of the Tuscaloosa, Anniston, Gadsden, Decatur, Montgomery metros, most of Huntsville's, and a third of The Shoals. Just the metropolitan areas combine for over 2.5 Million. That doesn't include the 5 or so micropolitan areas and a portion of Auburn's metro.
If we're talking about metro areas within driving distance (btw, pretty much only the first sentence of this post was pertaining to you're first paragraph), which would only be metros like Birmingham, Tuscaloosa, and possibly Anniston and Gadsden, you have 1.5 Million. Again, that's just the metros.
I agree with your last paragraph, though I don't know if Birmingham is so close to Nashville that it really affects its ability to attract a team. I'm from North Alabama, and I don't know a single Tennessee Titans fan back home. Sure, people would prefer that the Titans win than say the Bengals, but I seriously don't know anyone that actively follows them. I'd actually say that I know more Predators fans than I do Titan fans.
If you look at it this way, Nashville doesn't have an NBA team, and who really cares about the Hawks? That gives Birmingham a bit of an edge in that an NBA team would be the only game in town and would likely garner more support from those fans that are in a fuzzy, or gray, area regarding which city's basketball team they'd rather support.
BTW, all, a story on AL.com about it all. LINK