Whatever he does as leader, it won’t matter. This is a pyrrhic victory for the DUP’s membership - the mess has raised a risk of early Assembly Elections, which will mark the end of the DUP’s days in the limelight.
Arlene Foster was the moderate wing of the party (hardline unionist, but not a religious fundamentalist), and was chosen specifically to prevent moderate Unionist voters from defecting (back) towards the centre-ground UUP. A more even split of the Unionist vote between DUP and UUP will hand SF the role of largest party, and thus give them the dibs on the First Minister position. That’s the prize that has been lost last week.
Moderate Unionist voters vote for DUP rather than UUP because DUP was the only party that offered a chance of getting a Unionist as First Minister. If only the Deputy First Minister position is up for grabs from now on, then that block of voters will return to more moderate parties: no matter what way they vote, SF will get the First Minister job, so it’s better to stack the Unionist bloc of the Assembly with people who better represent their views.
In terms of powers, there’s no difference between First Minister and Deputy First Minister, but the First Minister is the one who shakes hands with the other heads of state, and does the high-profile figurehead work.