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FYI, I got it running. Here's how:

1. On download page, click "Get Steam key"; follow prompts to create Steam account, if needed; follow prompts to login to using steam username/password; return to Download and page click "Get Steam key". For me this produced a 15-character key of the form aaaaa-aaaaa-aaaaa.

2. Install steam-installer

$ sudo apt install steam-installer

3. Go to the application menu and start Steam (it was under "Games"; the first time Steam is run, it finishes installation and prompts you to login to a steam account)

4. In the Steam window, under Games, click "Activate a product on Steam"; enter the Steam key from the first step.

5. Click "play"; see that nothing happens; be confused; click all over the place; notice that the steam thingie thinks you're already playing the game when clearly you are not; close all windows; reboot computer.

6. Go to the application menu and start Empire Deluxe CE (it is now under "Games").

Note: at no point did I need to use the 77MB zip file I downloaded from the download page. I'm sure it is good for something. I just don't know what that thing is. Thanks for reading.