A triode vacuum tube has three electrodes, namely an anode, a cathode and a control grid in between of those. Assuming that the filament is hot, current would readily flow from the cathode into the anode when no bias voltage is applied to the grid, like a diode valve.
But doesn't the grid also have the ability to capture electrons? Would there be a non-negligible current path between the cathode and the grid?
The grid is maintained at a negative voltage with respect to the cathode (similar to the operation of a N-channel JFET, or a depletion mode N-channel MOSFET), so electrons will be repelled by it. A result is that fewer electrons reach the anode, which has a positive voltage with respect to the cathode in order to attract electrons. Whatever small current flows in the grid is considered leakage.
But doesn't the grid also have the ability to capture electrons?
You are correct, it does and some non-negligible current can flow.
However, to get a current the electrons need to actually enter the grid.
If we look at the construction of a triode:
we can see that the grid is just a thin wire, so the chance of an electron hitting (touching) it is small. Most electrons will "miss", travel between the grid's wires and reach the anode.
Depending on the voltages at grid and anode, the "pull" (due to the electric field) on the electrons from the anode might be stronger preventing the electrons from entering the grid.
Apart from tube design specifics (see what is said about grid shapes in other answers), circuit design keeping the grid sufficiently biased negative does.
"Grid current" is a well known phenomenon with vacuum tubes biased well into the positive. Considered part of some power amplifier designs (especially transmitters); obviously, the previous stages need to be able to deal with having to feed that kind of nonlinear load. Considered an overload condition in other designs.
If the grid goes positive, it does act as an anode & you get grid current. In transmitter class C output stages, they used to use the grid as a rectifier to generate its own bias, which was stored on the coupling capacitor. If you run a valve with the grid floating, it will pick up some electrons due to the Edison Cold Plate Effect and bias itself partly off until there is enough negative voltage on the grid to repel any more electrons.
