There are a number of reasons, many of which have been listed here. The single biggest one is the small size of a guitar signal compared to a diode drop, or to the drop of two diodes in a full wave bridge.
You can, in theory, solve this with more gain, as you thought. It's just that you have to use so much gain that you run out of power supply voltage to do the gain. You have to make the signal so big that 1.4V of diode drop is negligible compared to the signal being rectified even after it's decayed by a lot. Otherwise you get both odder distortion than FWB gives by itself and the gating effect. The amount of total signal (not just gain!) you have to have before the FWB drop begins to be something that can be ignored for audible effects is quite large if you do it well - multiple tens of volts. The ear is good at picking out some things.
There are ways around this. One is to use a transformer to eliminate dependence on the pedal power supply. The first "Octavias" did this. The center tapped transformer is more important for cutting down on the forward drop you're trying to make negligible than for lowering parts count. Another is to use diodes already biased nearly to conduction. This concept usually requires a solid state phase splitter; it's a technique used in the Fender Blender, Armstrong Green Ringer, and Fox Tone Machine. The Univox Super Fuzz and its half-a-dozen brothers and cousins uses a phase splitter and a forward biased transistor base-emitter junction to do the same thing.
You can use a full wave bridge by converting the signal voltage to a signal current and then sensing the current and reconverting this to a signal voltage; that's what's going on in the EH-5950; using a part as the feedback of an opamp forces a current through the part, and the opamp makes up any voltage differences to make the proper current flow. I hadn't looked at the 5950, but now that I do, that's what's going on. It's funny - they could have achieved full wave rectification with the same number of opamps (2) and fewer diodes (2) and fewer resistors by using the classical "precision rectifier" circuit with two opamps. I can only guess that they were doing something else with the signals that made them want to use this funny floating FWR.
The classical two-opamp FWR has not received much use in pedals. I dimly remember Anderton using one, and John Hollis used one in his omnidrive. That's strange, because the two-opamp precision FWR does the job you want, down to millivolts, and doesn't have other oddities about it.