“I immediately imagine this as a ’90s Bullfrog god-sim like Theme Hospital”

Yes I imagined it as a sim, but not a "god-sim." in the sense that you have the same control as an actual manager would. Think less simcity and more monster rancher I guess? I realize the basics aren't anything innovative (though the idea came from an early idea of an arena shooter where you're the arena instead of the lone spaceship).

“Is there any particular way that you imagined its implementation?”

About as you guessed. Ever played monster rancher? The training parts of the game (somewhat analogous to the strategy parts in E:SESH) are all menu driven, but the battle is real-time action. It could very likely be web-based. So you'll spend the between-weeks organizing matches, dealing with hurdles, etc etc. And then enjoy/oversee the matches. Ideally you'll have to throw in some kind of surprise somewhat regularly so you'll be regularly involved.

“How would the game scale? How would you see the logical progression, and what point does the game simply run out of new stuff to throw at you?”

I forgot to mention, I'm of the philosophy that losing can be (but isn't always) fun. Play the politics too badly and you'll end up left for dead in an aqueduct. The game will pretty much progress with your funds, as you gain more wealth you can stage more exciting battles, etc. (which also serve as your reward for progressing ;). Cesar acts as your "end-game" because a failure to please him will lead to your political ruin (via some mildly-historically-accurate-and-mildly-comedic situations most likely) while success means yea, you can keep playing, but you've peaked.