The importance of splitting the positions of head of state and head of governmentWatch
Countries that unbundle the two positions seem to lack the same problem, although giving the headship of government to the head of the executive branch seems to augment the same branch at the expense of other branches, and most of their heads of state still retain reserve powers that will still drag them to the political arena.
The closest things to heads of state in "presidential" systems tend to the first ladies.
One upside to unbundling the two positions is that a head of government will no longer have to worry about juggling between administration and ceremonies.
Another upside to it is that it can lessen the toxic effect of political strife in governance and beyond, because the symbol of the country will no longer have to pick sides in any partisan struggle, rendering one safe from public backlash.
This will lead to what Walter Bagehot called the "dignified" and "efficient" aspects of statecraft.