If you see logic in the B-E-A-D-G-C part - each root being the dominant of the next - why is adding the dominant of B to the front of it a problem?
It's not a complete cycle of 5ths. That would be C-F-B♭-E♭-A♭-D♭-G♭/F♯-B-E-A-D-G-C.
Why does it sound good to jump straight from C to F♯? Mostly because of what happens afterwards. If you'd stuck on the F♯ it wouldn't have been BAD, but it wouldn't have been particularly functional. But the whole progression is supremely functional - you can't get much more functional than a 'cycle of 5ths' bass line!
This is a very common progression. Sometimes we soften the shock of the jump to F♯ by using the nearer-to-diatonic F♯m7♭5 chord.
It's also interesting to note that a ii-V-I progression is so strong that you can jump into one literally ANYWHERE. Which is how 'Giant Steps' gets away with it.