How to amplify RF signal to high enough level for a CMOS downconverting mixer?
Here's a schematic of a typical CMOS mixer circuit:



It's the classical "Gilbert" mixer.

Although not clear from this picture, the bottom NMOS is just for biasing,
you can view it as a DC current source. That makes the bottom half of this circuit identical to a standard differential pair.

The inputs of this differential pair are connected to the outputs of the LNA so this differential pair simply converts the RF (voltage) signal into a current signal.

That current (containing the RF signal) is then fed into the upper "business" part of the mixer, the actual switching happens here. For that those 4 NMOS in a row need to be switching on/off properly. Therefore the LO signal needs to be large enough. Do we need 700 mV (as you claim), in the order of Vt for that?

No we don't! As long as the switching transistors switch "enough" then we will get an IF output signal. If the switching NMOS have a large enough W/L then even a 200 mV LO signal could be all we need.

What matters is the difference in Vgs for each pair of NMOS. As long as one NMOS has a larger Vgs than the other NMOS and the difference is such that the current from the RF part below chooses one NMOS over the other, then the mixer will "mix".

This is the same principle as applying a bias to a transistor in an amplifier circuit, by applying a bias we overcome the Vt (700 mV) "dead zone" of the NMOS so it is not an issue.