Depends on what you mean by get the job done. (Rather which job)

If you are doing research into designing advanced transistors with new geometry or new materials (which is what I did my graduate research in), you would be using something like DFT (Density functional theory) for equilibrium analysis and NEGF, Huckel theory etc. for simulating current. These methods only realistically work on ~500-1000 atom systems, beyond which the simulation takes too long to run even on supercomputers (which is what i was using). I think GPUs here would be very useful, but there weren't any tools at the time that were seriously optimized for GPU. The codes I was using were SIESTA/TransSIESTA, Atomistix, QuantumEspresso and others.

For simulating multiple transistors, or transistors with a large geometry (for example 14nm gate length), you would use TCAD simulators that use FEM + measured parameters to simulate the transistors. The equations behind these are traditional semiconductor equations with a bunch of heuristics and curve fitting. The main tool I used was Sentaurus TCAD.

For simulating larger circuits, say a low-noise amplifier or maybe a small DAC, you would use tools like Cadence Virtuoso + the provided PDK from your foundry. The equations here are simpler than the ones used in Sentaurus and they are also calibrated to measurement.

Just as a note, GPU supported code just now seems to be coming on line with like Quantum Espresso GPU version getting an alpha in 2019. I think a large part of that is due to the 64 bit precision making the speed up not as great on GPUs (and to a large extent eliminating the usage of consumer level cards which are popular at the classical molecular dynamics software like GROMACS). Plus Guassian, while having support since Kepler days note that earlier cards didn't have the memory requirements (which also is a huge problem for consumer cards) making adoption slower, because of the added costs.

Thank you for all these pointers, this is really cool.

In particular, I had never heard of DFT ... really intesting and first time I am exposed to what feels like real "hands-on" QM (as opposed to the very stripped down systems one learns about in introductory QM textbooks).

Full disclosure, I don't work on processors but am in a tangentially related field.

However, I don't feel that you need to have an analytical solution to the Schrödinger equation. In fact even in chemistry we don't do analytical solutions instead using fancy basis sets which allow us to do approximations.

Regardless, I don't think even that is particularly necessary, as quantum at that level means you basically have some level of leakage where the electrons can just tunnel through the barrier created by the transistor when off. So if I had to guess most of it is just ways to rectify this leakage so it doesn't effect calculations, probably similar to a form of error correcting.

(This ignores that you may have to do some initial quantum calculations using Density Functional Theorem to get a guess at how much leakage based off the materials you are using, though if I had to guess most of that work was done a while ago.)

Solid State Physics has a bunch of different types of models to try to explain the behavior of electrons in semiconductors and conductors. Most of them are only valid under specific situations and they sometimes give erroneous results unless you apply them carefully. That being said, your absolutely right that the industry uses approximations. They usually use some kind of simulation physics package similar to the ones used by EE engineers when designing circuits.