How to properly analyse this simple circuit?
This is a linear circuit and you can apply the principle of superposition. For a linear circuit with multiple sources, you analyze the circuit as a sum of the output response for each source individually, with all others disabled. For a voltage source, disabling means set it to 0V, a grounded node. For a current source, disabling means to set it to an open circuit, a floating node.
In this case, analyze the circuit with V1 = 5V and V2=0V. Vout1 = 5V * (R2||R3)/(R1+(R2||R3)
Because R3 is so big compared to R2, Vout1 will be 2.5V.
Next, analyze the circuit with V1=0V and V2 = 10V. Vout2 = 10V *(R1||R3)/(R2+(R1||R3) = 5V
Vout total = Vout1 + Vout2 = 7.5V