2n7000 used for inductively coupled load modulation apparently latched
Your simulation has some problems.

First problem is the FET is wired directly across a voltage source which has zero output impedance, so no matter how well it is turned on it won't affect the voltage.

Second problem is there is no return path for the Gate drive. The entire circuit is floating, so it simply jumps up and down in time with the Gate drive and the FET never turns on. 

Third problem is 200us is not nearly enough time for the 10uF output capacitor to fully charge, so even with the other problems fixed you won't see the 'steady state' waveforms. 

Your schematic shows the circuit grounded at the load, so I added a ground there. To (roughly) simulate the source impedance I added 250&ohm; to V1. I also added 100&ohm; to the Gate drive signal to simulate the output impedance of the LTC6990. The DC output took a long time to stabilize so I reduced C7 to 1uF and changed the simulation time to 1ms. Here are the results:-



Zoomed in to show a few cycles of the last FET 'off' period.  



The FET body diode is turning on during each negative half-cycle, flattening the bottoms of the waveform. You can also see a reduced amplitude wave feeding through to the Gate via capacitances inside the FET.    

My source impedance choice was just a guess, and doesn't simulate the complex impedances typically found in rf circuits. If the Class E amp and coupling coils were also accurately simulated the waveforms should look closer to what you got in reality. 

So much for simulation, now the real problem - you can't use a MOSFET to modulate an AC waveform larger than ~0.6Vpk without distortion because the body diode will conduct during negative half-cycles. To fix this you could put two FETs in series 'back-to-back' with Drains and Gates connected together, then one FET's body diode will always be reverse biased. Also you should provide enough Gate drive voltage to ensure the FETs turn on fully. 3.3V is not enough for a 2N7000. 

You might also consider putting the FETs in series with the source instead of in parallel. Then they would disconnect the source when off, which could be more efficient than 'shorting it out'.