There are many different liquid rocket engine power cycles, forxing propellant into the combustion chamber with tank pressurisation, gas generators, and heat exchangers, amongst others. Instead of using the exhaust gas from a gas generator, could a fan fixed to the outside of a missile be used to power the turbo pumps to then pump the propellant, using the airflow around the actual rocket, much like a ram jet. This would of course mean that it would have to begin with some amount of velocity, but this could be added with just a small booster stage, just like a ram jet. Is this at all viable and does it have any advantages aside from maybe simplicity or is it just gimmicky?
It is always worth considering where the energy doing your work will come from. In this case the energy is coming from the airflow, with it being slowed relative to the rocket, which means increased drag that the engines must overcome. So this is doing much the same as a turbo pump but adding steps. In particular the outside airflow will be at a lower density and velocity than a turbo pump combustion flow so need to be larger and heavier for same power capture. There is also the complication that a turbo pump power ramps nicely with propellant flow rate whereas the fan power available will be constantly changing as speed increases and density decreases probably meaning you need variable pitch blades and some complex design to handle transonic and supersonic flow.
Something like this does show up as ram air turbines for power which might be useful for a re-usable rocket during the unpowered part of descent to run hydraulics for steering/landing gear etc and where extra drag is a nice bonus. If you want really high reliability and low cost you might also use venturis to provide small amounts of mechanical power, though believe they are subsonic only so not useful over much of a space crafts profile but notable for the extreme simplicity and would theoretically allow the airflow to directly pump a fluid with no moving parts.
You also see ram turbine powered systems on aircraft where a strap on system needs power but do not have time to engineer connections to the parent crafts systems (example drop tanks and equipment pods), but one hopes you do not need to fit a jammer or drop tank to a rocket without doing the relevant engineering.