Summary: For people who experience the "game not starting" or "pressed play and nothing happened" issue, this is how I solved it. The solution is at the end.
I installed ESO on my Windows 10 system directly from my ESO account (non-Steam) and was unable to get the game to launch. The launcher started up and I clicked the "Play" button, the "Play Button" became grayed out, my cursor turned into a spinning blue circle (the "busy" cursor in Windows 10), and after about 3s or so the cursor returned to the normal pointer arrow and nothing else happened. The ESO process (eso64.exe) was running, but it was displayed in the "Background Processes" list in task manager. The launcher's "Play" button remained disabled (gray).
I pulled out my hair for a few days trying to solve this, running through multiple reinstalls, reboots, driver updates, Windows updates, etc. I downloaded SysInternals and tried investigating with ProcMon and Process Explorer. I read host files. I tried to run EasyFixes from Microsoft and then resorted to hacking the registry by hand. Nothing worked. Then, I guess I'd suffered enough because my Google-fu finally kicked up an idea: Disable all network adapters when starting the game and enable them after the ESO process is running. IT WORKED, for no reason that I can determine.
So, here is a summary of what works for me in the hopes that it helps some other poor slob who might be pulling their hair out:
1. Ensure networking is enabled.
2. Run the ESO launcher.
3. When the "Play" button is visible and enabled, disable all network adapters (Win key -> "Network Connections" -> right-click on each network adapter and select "disable").
4. Click the "Play" button in the launcher.
5. Wait for 15s (just because).
6. Re-enable your relevant network adapters (right-click on the adapter(s) in the "Network Connections" window and select "enable" this time).
7. Within 5s or so the screen goes black and the startup movies begin to play
I don't know why this works, but because of the plethora of strategies that are recommended for solving this problem that did not work for me, I thought it would be worth recording what actually did. Good luck, and happy hunting.