Existence and uniqueness of solutions to Thévenin circuits
If the circuit is made up of ideal voltage sources, current sources, and resistors, then there will always be a unique solution. As you have said yourself, it creates a system of linear equations that can always be solved.
The example you give for shorting the voltage source with an ideal wire does not make sense as you will have conflicting definitions of the voltage at across/at the node. The voltage source defines a voltage (i.e. 1V) across it while the wire will define no voltage (i.e. 0V) across it. To give an example to you with a math background, this is equivalent as saying 1=0 which is just not true.