How would a country be without leaders?
Anarchy. The country wouldn't be a country anymore. It would be a free for all, and an absolute disaster. Bandits would get rich, civilians would get killed and robbed at a constant rate. It would develop into a Fallout 4 kind of world. Or, more realistically, a middle east scenario. I think other nations would claim the land, easily taking us over, and becoming the most powerful nation.
The idea of countries without leaders is compelling. However, the challenge with the missing country leader might be the lack of vision.
Trash might still being collected and schools might still be running - or not. There are examples for both: Belgium did quite well for a while without government, other countries did suffer. There is high risk that over time the lack of vision and strong individuals will lead to deterioration.
The 'winning' nations in the world are those with strong leadership. Major reforms and achievements - historic examples are China, Russia or Turkey - are often driven by individuals, not by parties or committees. This does not mean that there aren't strong and capable teams and advisors behind the successful leaders.
The success of a country or organization depends on leadership. That means on the ability to place the right person at the right position at the right time.