In a studio you'll want a microphone that is as neutral sounding as possible.
For live, I'd say it's not always the case. Hence using bass drum microphones. They'll get the bass more than a normal vocal mic will. Your DI is going to be getting the neutral signal, the cab microphone is there specifically to colour the sound with the tone of the rig and cab, and a scooped mic will pick up those extremities more.
I've worked with enginerds that have been so picky with their microphones, using KSM9s for their vocals and shunning the SM58 completely. Yet they were still using a D112 or e602 for bass guitar.