Tangential question: Does anyone know of a basic large-signal equation for a triode (or any other vacuum tube type) like the simplified Ebers-Moll equation for BJTs or the square law equations for the linear and saturation regions of a MOSFET? It would really help my understanding, but whenever I google it I only see academic papers, like it's a weird thing to search for.

Vladimirescu, Andrei. The SPICE Book. John Wiley & Sons, 1994.

Gives overview equations for MOSFET device simulations which are probably sufficient for most purposes in Section 3.5, and COMPLETE mathematical descriptions of the SPICE MOSFET implementation in Appendix A.3. Not for the weak.

"All models are wrong, but some are useful." -- George Box

With that said, a N type JFET is not a bad start. The main rules of thumb work: The grid draws negligible current. The tube will pass enough current from plate to cathode, to maintain a roughly constant cathode voltage above the grid.

If I understand them correctly, Ebers-Moll equations are based on the exponential relationship between voltage and current in a BJT.

But tubes aren't current amplifiers, they're voltage amplifiers, like FETs.

You can look at the "characteristics curves" of tubes (plate curves and transconductance curves), which tell the story of current against plate-to-cathode voltages for fixed grid voltages.

The Koren equations are the only thing I've found.

"Improved vacuum tube models for SPICE simulations" https://normankoren.com/Audio/Tubemodspice_article.html

The intractability of the Triode is part of the reason why the Pentode exists. And, you will note, the Pentode curves in certain modes looks a lot like your bog standard MOSFET.

This also discusses how the "constants" ... well, aren't. https://www.john-a-harper.com/tubes201/