This guess is based on the second hint:
I don't start the fight -but I finish it.
I take a side-view approach to it all.
Yet when I work, the best will benefit.
Just stay focused, drawn by my silent call.
The checkered flag is raised at the end of a race ("fight"), not the beginning.
The checkered flag is located to the side of the finish line.
The checkered flag is raised for the winner ("the best").
You need to stay focused to win a race; the flag is silent.
When I get up, everyone pursues me.
When I lie down, I am still remembered.
My presence is thrilling -by some degree.
The group of fighters is ever membered.
When the checkered flag is visible (during the last lap?) everyone tries to get there first.
Even after the race is over, the winner/finish is remembered.
The finish of the race is the most trilling part.
The racers are remembered?
Try as I might, I have little power;
So many like me can run the nations.
But right here, it is always my hour.
All come to me, restraining all patience.
The checkered flag has no power, unlike other flags (caution flag, etc).
There are other flags such as each nation's flag.
At the race track, there is no other objective.
During a race, everyone races to the checkered flag as fast as they can (no patience).
They see me, my details are absolute.
When they catch me, they always overshoot.
The checkered flag is distinct from other flags.
When you finish a race, you overshoot the finish line.
Those who do not come to me praise those who do. They would not be "fighters".
The audience praises the racers.
To some, I look like a battlefield for monarchies.
The checkered flag looks like a chessboard.