What makes it possible for FETs to control current with voltage directly, but that bipolar transistors need to use current for control?

Well, it's not that bipolar transistors need to use current to control the current from collector to emitter; we just often to think of their use in that way (we could think of the voltage from base to emitter, or from collector to emitter, as a control, since $I_c$ does depend on both of those).

But the nature of the devices differ so that what we usually think of as a control quantity differs. For the bipolar transistor case, there is some current from the base into the device, and which $I_c$ vs $V_{ce}$ curve you are on depends on $I_b$-- so the $I_c$ saturation current depends on $I_b$. So we tend to think of $I_b$ as the quantity that we vary to get a desired $I_c$. Physically, the story is a bit complicated-- current flowing into the base means electrons pulled out of the base, which prevents the base from charging up and blocking collector-emitter current flow.

For the FET case, there is very little current into the transistor from the gate. For the JFET case, that's because the junction is reverse-biased; for MOSFETs, it's because the gate is actually insulated from the channel. $I_g$ is just tiny and not so relevant. In the FET case, the value of $V_{gs}$ is what determines which $I_d$ vs $V_{ds}$ curve you are on, and hence is what determines the saturation current (or channel resistance, in the resistive regime). So we tend to think of $V_{gs}$ as the control parameter. Physically, we can think of this bias voltage as creating an electric field that determines how easy it is for current to flow in the channel (that's the ``field effect'').