Realistically, a big part of the strength of the executive comes from presenting a united front across the various offices comprising it. For this to work, we'd probably need to make the executive far more powerful to compensate for losing that united front. President is already enough of a booby prize as is.
Because let's be honest... the Executive essentially serve at the President's pleasure. So you'd have someone in their team who they don't want and might actively antagonistic. Plus PiT is correct here that doing this at the same time as the President's powers continue to be whittled away is just a bit silly.
Why be President?