Have I made a gross mistake somewhere?

Well, I wouldn’t call it a “gross mistake”. What has happened is that you are overgeneralizing from a specific example to an incorrect general conclusion.

Your specific example is:

a rigid body system made up of three particles, initially at rest and positioned along a vertical line, with the middle particle separated by some distance d from the other two particles.

And your over generalized conclusion is:

for a rigid body to remain rigid, Newton's Third Law cannot hold in its strong form

This conclusion is incorrect, or at least it is not generally correct. For your specific example it does hold, but not in general.

In the strong form of Newton’s 3rd law the forces between a pair of point particles must be a central force. A stiff central force then is a central force which preserves the distance, but not the angle between particles.

The particles act as hinges between stiff springs. Your configuration is a hinge between two stiff springs, which is clearly not a rigid structure. However, if you arranged the three particles into an equilateral triangle then the arrangement becomes rigid. You can push on any hinge and they will all move together.

So your conclusion should be that idealized Newtonian rigid bodies cannot be modeled as a straight line of point particles interacting through a stiff central force.

Have I made a gross mistake somewhere?

Well, I wouldn’t call it a “gross mistake”. What has happened is that you are overgeneralizing from a specific example to an incorrect general conclusion.

Your specific example is:

a rigid body system made up of three particles, initially at rest and positioned along a vertical line, with the middle particle separated by some distance d from the other two particles.

And your over generalized conclusion is:

for a rigid body to remain rigid, Newton's Third Law cannot hold in its strong form

This conclusion is incorrect, or at least it is not generally correct. For your specific example it does hold, but not in general.

In the strong form of Newton’s 3rd law the forces between a pair of point particles must be a central force. A stiff central force then is a central force which preserves the distance, but not the angle between particles.

The particles act as hinges between stiff springs. Your configuration is a hinge between two stiff springs, which is clearly not a rigid structure. However, if you arranged the three particles into a triangle then the arrangement becomes rigid. You can push on any hinge and they will all move together rigidly.

So your conclusion should be that idealized Newtonian rigid bodies cannot be modeled as a straight line of point particles interacting through a stiff central force.

Have I made a gross mistake somewhere?

Well, I wouldn’t call it a “gross mistake”. What has happened is that you are overgeneralizing from a specific example to an incorrect general conclusion.

Your specific example is:

a rigid body system made up of three particles, initially at rest and positioned along a vertical line, with the middle particle separated by some distance d from the other two particles.

And your over generalized conclusion is:

for a rigid body to remain rigid, Newton's Third Law cannot hold in its strong form

This conclusion is incorrect, or at least it is not generally correct. For your specific example it does hold, but not in general.

In the strong form of Newton’s 3rd law the forces between a pair of point particles must be a central force. A stiff central force then is a central force which preserves the distance, but not the angle between particles.

The particles act as hinges between stiff springs. Your configuration is a hinge between two stiff springs, which is clearly not a rigid structure. However, if you arranged the three particles into an equilateral triangle then the arrangement becomes rigid. You can push on any hinge and they will all move together.

So your conclusion should be that rigid objects cannot consist of a straight line of point particles interacting with a central force.

Have I made a gross mistake somewhere?

Well, I wouldn’t call it a “gross mistake”. What has happened is that you are overgeneralizing from a specific example to an incorrect general conclusion.

Your specific example is:

a rigid body system made up of three particles, initially at rest and positioned along a vertical line, with the middle particle separated by some distance d from the other two particles.

And your over generalized conclusion is:

for a rigid body to remain rigid, Newton's Third Law cannot hold in its strong form

This conclusion is incorrect, or at least it is not generally correct. For your specific example it does hold, but not in general.

In the strong form of Newton’s 3rd law the forces between a pair of point particles must be a central force. A stiff central force then is a central force which preserves the distance, but not the angle between particles.

The particles act as hinges between stiff springs. Your configuration is a hinge between two stiff springs, which is clearly not a rigid structure. However, if you arranged the three particles into an equilateral triangle then the arrangement becomes rigid. You can push on any hinge and they will all move together.

So your conclusion should be that idealized Newtonian rigid bodies cannot be modeled as a straight line of point particles interacting through a stiff central force.