Cathode emission is relatively weak. A typical sweep tube needs 15W of heat for about an ampere of emission, and that's supposed to be pulsed at that; a bigger transmitter tube requires hundreds or thousands of watts.

And yeah, between the space charge neutralization property of a plasma, and the much greater emission due to ion bombardment, the current density can be much higher.

We can go even further and consider a stationary matrix of ions; this further reduces the energy levels required (~eV to reach conduction band, versus ~10s eV to reach vacuum state), and greatly increases the current density. Unfortunately, transit time is significantly impaired, because transport is via diffusion, at quite low velocities considering the energy levels and an electron's [free] rest mass). On the upside, the breakdown voltage is also enhanced, so we can make very thin junctions, and this more than makes up for transport.

In other words, from a sufficiently high level, we can look at tubes and transistors as the same general thing: some exchange of charge carriers at the electrodes, with transportation inbetween, subject to applied fields. For tubes, we use the ballistic transport of electrons, and space charges; for transistors, we use diffusive transport, and recombination. Solving the equations gives a V^(3/2) characteristic for tubes, or a exp(V) characteristic for transistors. (Depending on large/small signal regime. Tubes, FETs and BJTs are all exponential at low current densities, driven by the tail of Maxwell-Boltzmann statistics. At higher current densities, tubes go towards V^(3/2), FETs go towards V^2, and BJTs remain ~exponential.)

Of course if we put in a plasma, we get all kinds of weirdness, typically negative resistance and bistability. This happens because emission is proportional to current flow times a multiplier (an ion impact begets multiple free electrons, on average). The plasma is also not easily controlled by fields: due to its being conductive, it's an effective [electric] shield. (The same is of course true of semiconductors, but we can harness very large surface areas in that case, making FETs practical.)

Tim