The opposite of "hand egg" is not "foot egg".

And the point is that the object played in "American Football" is not a ball. Balls are round. It is egg shaped. The object played in "Football" is a ball. Describing the ball in football as an egg just makes you look like you can't see properly.

that just sounds petty for the sake of pettiness, surely you can do better in terms of mocking

Which is foot ball. Not foot egg.

Soccer is also known as "divegrass", not footegg...

The opposite would be foot ball. Although both are petty as anyway both are called it because it is played on foot with a ball. Not because you use your foot to kick a ball.