Why does my non latching relay design not switch back to original state as I expect?
This design looks correct, so it should be working.

From what you are describing, it looks like relay driver (MOSFET and protection diode) is fine. If you see 0 volts on both coil pins 1 and 12, and your relay is engaged (pins 5 and 4 connection), this means you have damaged the relay.

The relays usually get damaged this way because you try to pass to much current through them. I think the reason for that is inrush current. Many electronic devices have big capacitors on input, which charge very quickly when the power is applied. During this time, capacitors have effectively 0 ohm resistance, so the maximum charging current is limited by power supply, wires, and stray inductances. The result is very short, very high current peak. It can be easily many tens of amps, and your relay is rated for 2 amp max.

Note that inrush current is very short, so you cannot measure it with regular voltmeter -- you need an oscilloscope measuring the shunt.

The easiest way to fix it is to oversize relay and add some resistance. For example, if you add 1 ohm resistance on the input, you'll limit inrush current to (12 volt)/(1 ohm) = 12 amperes in the worst case. JS1-12V-F relay is rated for 10 amps and will likely work fine. Another downside is that resistor will waste some voltage -- so with 0.5 amp load, your device will only get (12 volt - 0.5 amp * 1 ohm) = 11.5 volt, and the resistor will emit (0.5 amp)**2 * 1 ohm = 0.25 watts of power as a heat.

There are better ways to decrease inrush current, like pre-charging and NTC thermistors -- they might allow you to use smaller relay, but would be harder to design.