When I tried installing Warframe on SteamDeck, I encountered this problem. It appeared to download several GBs of content for some time and run Vulcan shaders. However, when the installation finished, the game didn't launch.
When I checked the game directory (~/.local/share/Steam/steamapps/common/Warframe/) it was empty. When the game was launched in Desktop mode, it complained ~/.local/share/Steam/steamapps/common/Warframe/Tools/launcher.exe was missing. (Of course, because the directory is empty.)
Here's how I fixed it and got Warframe to work on SteamDeck:
I added the game to my library on Epic Games Store. Then, downloaded Warframe using the Heroic Games Launcher. But note that I didn't launch it. (I wanted to run the steam version of the game, not Epic version). Heroic had downloaded several MBs and now there was a Warframe.zip inside ~/Games/Heroic/Warframe/. I then extracted that zip into ~/.local/share/Steam/steamapps/common/Warframe/. Voila! now there's tools/launcher.exe. When you launch the game on Steam now it will start.
The game is very optimized. It ran smoothly. The only minor annoyance is that I have to log in to Warframe account in the launcher every time.