Is there current flowing through this blank wire?
My guess is no. Because both of those points are connected to a ground symbol by blank wires, they both have potential of 0V.


Your guess is right, but your reasoning is incorrect.

In any idealized circuit like this, the potential is equal at all points on the wire, but we still allow that current can flow. Because the wire is modeled to have 0 resistance, it doesn't require any potential difference across it to establish a current.

The reason there is no current in this wire is because of the cut-set form of KCL. If you were to break that wire, there'd be no other connection between the two halves of the circuit. Therefore no complete circuit can form using that wire, and so we know no current is flowing through it even when it isn't broken.