How much role does radioactivity play in making Earth geologically active
The idea that the heat of the earth is from the time of its formation dates back to the late 1800s before radioactivity was known.  Unfortunately for that theory it was shown that an initial blob of molten rock would cool to a solid chunk, radiating its heat into space, in a few tens of millions of years.
When radioactivity was discovered it was seen as a way out of the problem but the known radioactive elements were sufficiently rare in the earth's makeup that they couldn't provide enough heat.  Finally in the fifties the bulk composition  of the earth and the occurrence of radioactive isotopes were well enough known that it was shown that there is enough radioactive potassium, with a long enough half life, to provide the observed heat flux from the earth's interior.  Most of the heat is generated in the mantle, not the core since, as you point out, that is where the potassium is.  The mantle is also the layer that has convection currents in it to release that heat and drive plate tectonics.
The outer core is liquid because, at the pressure it is under,  being a liquid is the thermodynamically stable state of the heavy core minerals.  The ice under a skater's blade turns into a thin layer of water under the passing skater's pressure, lubricating the passage of the skate.  The water does not warm up much, it is still below freezing, but it is not stable as a solid under the pressure of the skate. As soon as the pressure is released the water returns to ice which is the stable state of H2O at atmospheric pressure and sub-freezing temperatures.  Similarly the outer core is liquid because that is its stable state given the pressure and temperature at that depth.
Increase the pressure more, as happens as you go deeper, and the stable state of the core material is a solid.  This explains the reason the inner core is solid.