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No real expertise here in electronics, but 15 years or so ago I did a fairly complete literature survey dating back to the early telecom studies from the 30's on. There are a couple of issues with diode mixers:
One is pretty easily dismissed: When the diodes are either on or off, there is significant current thru the diodes and the small signal current adding and subtracting from the LO current is typically not significant.

During switching you want as fast a rise and fall time as possible as the signal voltage does modulate the switching current in that interval. You want as high switching power as the diodes will tolerate to change the (sinusoidal) voltage as fast as possible to get out of that interval. The diode then saturates and all is well. That is why the higher level mixers have diodes in series and also resistors to limit the current during saturation.

Now at low frequencies, our faster digital circuits have fast switching times and operate from low voltages to saturation. Obviously at VHF, the transition time becomes a factor, not the on and off times until the waveform begins to look like a sine wave.

But I think there is an added factor. With the FET or switch type mixers, the LO current is not in the signal path. Or is it reflected in the turn on/off resistance change? Off hand would think this is a different breed of cat.

The real alternative is the old beam deflection mixer such as the 7360 where the signal current really never changes and is beam deflected from one plate to the other with almost no change in beam current. Of course, it has other problems, can you say microphonics and heat.

Someone straighten me out as needed.