I just watched the old news coverage. I misremembered the Bush is MIA stuff, they said in order
- Reagan is fine and was not hit
- Actually he was hit under the arm and may need surgery but it didn't hit his heart and he's in no danger
- The Vice President had been in Texas and was en route somewhere ("we're not sure what was next on his itinerary but regardless" he's diverted back to DC and we don't know when he'll arrive
- Actually, it is in fact a chest wound and he got a transfusion and they are doing surgery now but we don't know how serious it is
- All of the key cabinet personnel are heading to the hospital now
- Haig did a presser and said "I've been in contact with the VP..." and "I am in control at the White House"
That's what unfolded on live TV. So Bush wasn't MIA, he was just invisible as the President's condition was seemingly getting worse. Furthermore, the White House apparantly had worse info than the press as the Press Secretary didn't know Reagan had been taken into surgery, and the press was grilling him like "is all your information going to be this late?" It was a pretty uncertain time.
Bottom line: Haig was trying to reassure people that the country had leadership, that we weren't adrift. Bush couldn't. Reagan couldn't. Haig did. He wasn't trying to claim power.