Power supply problem, what is the correct solution?
So, the question is whether the supply is operating correctly or not.
A practical answer:
The LM7812CT requires a voltage difference of at least 2VDC across its input and output terminals. This voltage is called Dropout Voltage. So the minimum input voltage for proper operation should be \$\mathrm{V_O+V_{DROP}=12+2=14V}\$. Since the input voltage comes from a bridge rectifier+smoothing capacitor, the valley of the ripple should not be below 14VDC.
According to the given info in the question, the valley of the ripple is \$\mathrm{V_{in-min}=V_{DC} - (V_{rpp}/2)=17.64-4.22/2=15.53V}\$. This is higher than the required minimum input voltage. So the answer is yes,  the supply is operating correctly.
A detailed answer:
The ripple factor formula you're using is a simplified form. When it comes to practice, there are some other things to consider like the ripple current flowing through the smoothing capacitor. Anyways, let's use the simplified formula:
$$\mathrm{V_{rpp}=\frac{I_{LOAD}}{2\cdot f_L\cdot C}}
$$
If the circuit is operating correctly, the output voltage should be pure 12VDC. So the output current is \$\mathrm{I_{O}=12V/20\Omega}=0.6A\$. Since 7812 is a linear regulator, its input and output currents are equal. Thus \$\mathrm{I_{LOAD}=I_O}\$ and \$\mathrm{V_{rpp}=0.6A/(2\cdot 50Hz\cdot 1000\mu F)=6Vpp}\$. The expected ripple voltage is 6V and the measured ripple voltage is 4.22V which is lower than expected. This is good. So, the answer is yes, the supply is operating correctly.