Thanks for the good clue, which led me to the recipe that works, which is slightly different:
In Steam, click “Steam” in far upper left corner. This pulls down a menu.
On that pull-down, click “Settings”. This pops up a window whose left panel is a menu titled “STEAM SETTINGS”.
On the STEAM SETTINGS menu, click “Compatibility”. On the right panel activated by that selection, activate the toggle “Enable Steam Play for all other titles”. That pops up a small “Restart Required” window. Choose “Restart” (of course!).
After the restart, navigate the menus again to “STEAM SETTINGS/Compatibility”. Beneath the “Enable Steam Play for all other titles” item, notice a new line titled “Run other titles with:” and a pull-down for choosing amongst versions of Proton.
I tried only the default, which is “Proton Experimental”.
Close the “STEAM SETTINGS” window and click LIBRARY on the main Steam menu
I already had EVE online in my library, so I chose it and clicked Install.
Download takes forever on my ADSL Internet service.
When download complete, click Play, and …
The EVEonline launcher now wants to download 4.3GB. This will take roughly forever on my Internet service.
But so far, so good!