I would have thought that the 5V signal on the FET will not work. The FET is up at 4.7V so you don't have any headroom to turn it on unless your reference is with respect to the 4.7V zener.
I would look at a small 3 terminal regulator that could be configured as a constant current source and use the FET to connect it to 0V when you want it turned on.
The circuit will work as intended. It should be considered as a constant current generator circuit enabled by the mosfet. I've attached the simulation screenshot (since I'm lazy for the NMOS I used the NDC7002N that is similar to the one chosen by the OP) and the LTSPICE files.
Best,
0xfede
The zener diode isn't doing anything. Look at the voltage across it and current flowing through it. It's way below the zener turn on/threshold voltage.
If the 5V logic supply is regulated, then remove the MOSFET and zener and it will work perfectly. Otherwise, use a lower voltage zener, with a suitable resistor, powered from the 5V control signal.
By the way, just upload the .asc file. No need for the other files in a zip.