Can't play jazz for shit.
The word 'blues' is too broad a term anyway. Blues has a of genres in it and also regional scenes that had their own sound (and could also be considered, to some extent, sub genres of blues too)
You say B.B.King, Clapton and SRV, but it wouldn't make you a bad blues player if you didn't have a grasp on all three styles. B.B.King's style if radically different to SRV's, even if it's still, blues.
If I were talk about my abilities as a blues player, I see myself firmly rooted in that more rock-y style of SRV, The Hendrix Experience, Led Zep, Tony Iommi etc.
Although I can 'shred' my soloing is typically still that blues rock style.
Hell, look at Joe Satriani. The guy is equally good at blues as the top bluesman, and shits all over most so called 'blues' players you'd see at the local bar/pub. Develop that feel for like, like he did, and you can definitely do it.
One good example I like to watch is the Red House performance on the first G3 DVD that came out, with Eric Johnson, Vai and Satch.
Vai just seems out of place and doesn't seem to really understand the blues well at all, but Satch pulls out authentic sounding Albert King and Hendrix style licks, while also retaining a style of his own.