If I remember correctly, part of the reason for sending the Stones here was so that they couldn't misbehave quite as much and would actually finish their music.
This is part of the lore they told us when we did the tour at the studio a couple of years back. Not only was there almost nothing to do around there but record music, it was also a dry county. Of course they had their ways of scoring booze (and other things) anyhow, and there's even a "secret" room with a bar in the basement of the studio, but there were no clubs or the likes that they could frequent.
Moreover, very few locals actually knew them at the time. Everyone knows The Rolling Stones these days, but back then without the internet word didn't travel as fast, and very few people in Muscle Shoals or Florence at the time had even heard of the Rolling Stones let alone seen them. Apparently this was true for most artists who recorded at the studio back then. So aside from there being few distractions, the talent could actually roam free in town without being bothered by photographers or locals.
It seems to me that there was this perfect confluence of circumstances that made muscle shoals sound studio such a great place to record at the time. It wasn't far from Memphis or Nashville, while also being private, more or less distraction free, great studio musicians and engineers and perhaps most importantly a great sound.
Another cool story they told us was when the stones where there to record they didn't have all the lyrics to wild horses ready, or they had to redo parts of it, so Keith Richards (if I recall correctly) locked himself in the bathroom and didn't come out till they were done. The rest of them impatiently hanging out right outside waiting for the man to emerge with a hit. They referred to the toilet as the most important song writing room in the house. :o)
When we were there one of my buddies got so inspired by the studio that he actually came back about six months later or so and recorded with a band we'd met in Jackson the night before. Sadly the video is in black and white, but you can at least get a bit more of a sense of space in the studio than from the pictures in the article: https://youtu.be/s_HHgeKLqog
It's a wonderful place to visit, highly recommend it.
Well, if you look at a map, Muscle Shoals is fairly close to equidistant to Memphis, Nashville, and Birmingham. (Although, as I recall, it might as well have been Antarctica because nobody in the Rolling Stones could drive a car ... LOL).
However, it's probably mostly happenstance relating to a laissez-faire (for the time) attitude toward race from the area. WLAY would play both "hillbilly" and "race" music. Sam Philips (founder of SUN Records and ex-DJ from WLAY) and Rick Hall (founder of FAME Studios) were quite happy to record (some people would say exploit) artists regardless of race and that was a really big deal at that point.
"You'd go into a town and you'd find somebody nice enough to fix a dinner for you and let you sleep in a bed," Rush said. "You'd put some mattresses on the floor ... or you'd sleep in your car. That's what we had, man, and we didn't think nothin' about it."
And, do remember, The Blues in the US was almost dead by 1965--rock had pushed everything aside. It was all those silly Brits with electric guitars picking it back up and putting it into rock that effectively resurrected it.
Translated to: nothing else to do.
If I remember correctly, part of the reason for sending the Stones here was so that they couldn't misbehave quite as much and would actually finish their music.