Notice how none of those flat earth maps of the eclipse show those southward-going eclipses that happen over Australia.
The conditions are not always the same. YES the moon is always going north south or south north during these eclipses, but that is only one variable. You must expand your mind so that you can think of more than one variable at a time. As hard as that may sound to you, it's necessary.
Other variables include tilt of earth relative to sun, which changes throughout the year, and North-South position of the moon relative to the plane of earth's orbit.
Earth's tilt is such that at some points of the year, the north pole is more pointed towards the sun, other points the south pole, and other points the tilt isn't pointing to the sun at all. You haven't thought about that, but it's a variable you must be able to imagine. All of us globies can imagine it.
And the moon shadow isn't always starting on the very-north side of earth, or the very-south side. It's clearly, obviously, sometimes starting more central.
So these are two variables you have not been considering that will affect the shape of the path of the shadow.