If you note I said Head of State not Head of Government. In many Parliamentary systems the Prime Minister is the head of Government not the Head of State.
But we could spent hour discussing the pros and cons of Parliamnetary vs Presidential systems.
And this thread is about the Electorial College vs popular vote to elect the US President.
The head of state is problematic because often they are little more than figureheads, such as the President of Germany or the Governor General of Canada (if he's even a head of state in some form).
Anyway, getting back to the electoral college, there is no right to vote for President in the Constitution, but we do so because it's a more accurate way to determine the public's will. It's essentially a poll, and like all such measures has some inherent weaknesses.
One benefit of the electoral college is that it recognizes that the voting is a poll, and allows us to break the polls down by state, adjusting for population. This can be very useful. For example, what if hurricane Sandy hit a week later than it did, pounding the North East right on election day, trapping people in their houses and closing most of the voting precincts?
In a popular vote it might cost Obama millions, if not tens of millions of votes. But since we've broken the voting down by region, the electoral college vote would be barely effected, if at all. If only three people in New York made it to the polls, and two of them were Democrats, the choice of the people of New York would still be accurately reflected, which is all we actually need.
As an aside, the reason we adopted using a head count to determine a course of action, or who rules, is that changes in combat tactics (Swiss pikemen) made it obvious that whoever has the most heads wins. So instead of actually fighting a battle, they could just count the numbers on each side and see how the hypothetical battle would've turned out anyway, eliminating the needs to actually fight. In the early incarnation you had to show up with a pike to have your vote count.
Following that logic, one of the reasons I'd argue against mandatory voting is that neither side should be able to claim people who aren't going to show up to fight, but of course if actually pursued logically it would throw us back to some sort of Heinlein "only soldiers vote" system long after soldiering has become a very narrow and specialized exercize.