So why isn't the projectile changing speed at its peak?

A previous post posits a question, and a student's response.  The response seems okay, because every sentence is in fact correct.  But a logical connection isn't there.  Today I'll explain the error in reasoning, and give the better answer.

See the previous post here.  It discusses how to handle the conversation with the student who knows that they're right, even though they're wrong and you can't quite articulate the error on the spot.  That was my purpose in posting this question, as I had heard quite a bit of debate in the physics teaching zeitgeist about this problem.  I'm only posting the solution separately so as not to distract from the pedagogical message of the first post.

Here's the question:

A projectile is thrown upward at an angle.  At the instant the projectile is at the peak of its flight, is the projectile gaining speed, losing speed, or neither?  Justify your answer.

A student responds, "The path followed by a projectile is a parabola.  At the peak of motion, the slope of that parabola is flat, or zero -  the change in speed is zero.  The projectile neither gains or loses speed."

What's wrong?

The student connects the slope of the projectile's path to its speed.  That's not right.  The slope of a trajectory - the path that an object takes - says nothing about speed, just about the direction of motion at that instant.  [The student may well be confusing the slope of the projectile's path with the slope of a position-time graph, which does indicate velocity.]

What's the better answer?

At that instant, the acceleration is downward, and the velocity is horizontal - thus there's no component of acceleration in or opposite the direction of velocity, so the object cannot speed up or slow down.

Or:

At that instant, the net force on the object is downward, and the instantaneous velocity is horizontal - thus there is no work done by the net force, and so the object's kinetic energy cannot change.