Fictitious forces naturally arise in non-inertial (non-accelerating) reference frames and you have to be careful with them. In this example, it only leads to confusion.

$F = ma$ tells us that when the (total) force is zero, the object will continue in a straight line. It's not the centripetal force, but the absence of a centripetal force that makes the object fly away.

Centripetal force does come with an 'equal and opposite force' - in your example, you have to exert a force to keep your end of the rope in the centre of the rotation.

Fictitious forces naturally arise in non-inertial (accelerating) reference frames and you have to be careful with them. In this example, it only leads to confusion.

$F = ma$ tells us that when the (total) force is zero, the object will continue in a straight line. It's not the centripetal force, but the absence of a centripetal force that makes the object fly away.

Centripetal force does come with an 'equal and opposite force' - in your example, you have to exert a force to keep your end of the rope in the centre of the rotation.