... although I'm sure the philosophical approaches of both vary more by club than anything else....
Can't agree more. The club where I have been training BJJ for the past 8 years has a very favorable look on Aikido. The stand up techniques we learn to apply for self defense are often ikkio, nikkio, sankyo or kote-gaeshi (sorry if mangled the spelling). My instructor doesn't often use Aikido lingo, but teaches it as "defense against a one handed lapel grab, variation #1, and variation #2 " This is how I learned it.
A few years ago, my work scheduled changed for a few months and I couldn't train with him regularly, I joined an Aikido dojo for that time and was like, "Cool, I kind of know this". I say kind of because our stand up classes were maybe once a month and there is a certain amount of use it or lose it with my muscle memory.
After getting back to my regular club and talking to my instructor about it he told me that "Yeah, you can call it Aikido if you like. I prefer to call it Jiu Jitsu. Wrestling, Boxing, Karate, call it whatever you want, but to me they are all different sections of the same sphere of movement I like to call Jiu Jitsu." He told me another time, "Brazilian Jiu Jitsu, just takes Aikido and puts it on the floor. Its the same concepts of spherical movement, blending and redirecting our opponent. In bjj we just use our backs and hips instead of our feet and hips to generate the movement. (think lying down vs standing up) He holds shodan in aikido and judo, nidan in tang soo do and has many years in freestyle/folkstyle wrestling, thai boxing and escrima apart from his bjj training.
Personally, I like his philosophy. Call it what you want its all jiu jitsu to me.
...Aikido is billed as a generally useless martial art for self defense (other folk's words - not mine) while BJJ and similar styles get the media blitz ....
First see above about self defense.
To address uselessness, When sparring 1 on 1 my instructor will almost always use the classic MMA (thai box/bjj). As soon as another person enters the mix its an aikido game. The reason being not that Aikido is useless for handling 1 on 1, but Aikido becomes the only option (in his opinion) for handling more than 1.
Thanks for reading.