Perhaps this story might be sufficient which uses steam-login which uses xfwm4 under the covers to provide a lightweight desktop with Steam on top.

Here's the information from the README:

sudo add-apt-repository ppa:thor27-gmail/steam-desktop 
sudo apt-get update 
sudo apt-get install steam-login

Then logout and back in to get a Steam session. See also:

• What are PPAs and how do I use them?
• Are PPA's safe to add to my system and what are some "red flags" to watch out for?

• Just copy the steam.desktop file from /usr/share/applications/steam.desktop to /usr/share/xsessions.

• Then edit it and change the Exec option to

  Exec=steam steam://open/bigpicture
  

  or just create you own:

  [Desktop Entry]
  X-SuSE-translate=true
  Encoding=UTF-8
  Type=XSession
  Exec=steam steam://open/bigpicture
  TryExec=steam steam://open/bigpicture
  Name=Steam
  Comment=Steam
  

I have actually created a project to do exactly what you're describing here:

https://github.com/ShadowApex/steamos-ubuntu

It is just a small bash script that will do the following things:

• Create the steam user account if it does not exist.
• Install steam, if it is not installed.
• Install the Steam Compositor, Steam Mode Switch, and boot splash themes.
• Configure autologin for the steam user account.
• Configure the default session to the Steam Compositor.
• Create reboot-to-[steamos,desktop]-mode scripts to switch between sessions.

I've only tested it on a cleanly installed version of Ubuntu 18.04.

Another answer references:

sudo add-apt-repository ppa:thor27-gmail/steam-desktop

But from the author's Steam Login github page it is stated:

The PPA is really outdated, and I'm not much inclined to maitain it anymore. If anything changes I will update here.

By comparison the github page was modified in late 2017.

Arch Linux

steam

Arch Linux has some Tips and Tricks for running steam:

Big Picture Mode without a window manager

To start Steam in Big Picture Mode from a Display manager, create a /usr/share/xsessions/steam-big-picture.desktop file with the following contents:

[Desktop Entry]
Name=Steam Big Picture Mode
Comment=Start Steam in Big Picture Mode
Exec=/usr/bin/steam -bigpicture
TryExec=/usr/bin/steam
Icon=
Type=Application

Steam skins

The Steam interface can be customized using skins. Skins can overwrite interface-specific files in ~/.steam/root.

To install a skin:

• Place its directory in ~/.steam/root/skins.
• Open Steam > Settings > Interface and select it.
• Restart Steam.

An extensive list of skins can be found in this Steam forums post. Note: Using an outdated skin may cause visual errors.

Creating skins

Nearly all Steam styles are defined in ~/.steam/root/resource/styles/steam.styles (the file is over 3,500 lines long). For a skin to be recognized it needs its own resource/styles/steam.styles. When a Steam update changes the official steam.styles your skin may become outdated, potentially resulting in visual errors.

See ~/.steam/root/skins/skins_readme.txt for a primer on how to create skins.

General Gaming Advise in Arch

It's quite a lengthy page on Gaming covering both audio lag and video fps. There is even a section on creating a separate X-Windows environment just for gaming.

Double check your CPU frequency scaling settings

If your system is currently configured to properly insert its own cpu frequency scaling driver, the system sets the default governor to Ondemand. By default, this governor only adjusts the clock if the system is utilizing 95% of its CPU, and then only for a very short period of time. This saves power and reduces heat, but has a noticeable impact on performance. You can instead only have the system downclock when it is idle, by tuning the system governor. To do so, see Cpufrequtils#Tuning the ondemand governor.

(16.04 only, packages appear to be unmaintained).

I have found that the best way to do this is to use SteamOS's own packages for this:

sudo add-apt-repository ppa:mdeslaur/steamos
sudo apt-get update 
sudo apt-get install steamos-compositor steamos-modeswitch-inhibitor steamos-xpad-dkms plymouth-themes-steamos

The plymouth-themes-steamos installs the splash screen if you're building a dedicated gaming console, and the steamos-xpad-dkms brings in the controller fixes if you're using an XBox or Playstation controller.

You might want to turn on auto-login if you want a console-like experience:

• How do I enable auto-login in LightDM?

See also:

• What are PPAs and how do I use them?
• Are PPAs safe to add to my system and what are some "red flags" to watch out for?

Steam as program is running like a client and uses browser only for downloads.

When a game is installed, then you login only via browser before you start the game. After starting the game, you can minimize the browser or end the browser. Then you simply play the game.

To run steam as stand-alone product, you could install steam on to an external USB-harddisk. Before this you need to create an USB-stick with steam-files.

https://store.steampowered.com/steamos/buildyourown?l=english

For what it's worth, I don't think it is necessary to install anything. Or to have any desktop at all, lightweight or otherwise.

I'm assuming Steam is a game that requires graphics and hence x. You don't have to have a desktop to start x. You'll have to look a little to find the equivalent in whatever 'buntu you are using but in Lubuntu 13.04 to have an x session with no desktop you delete the line beginning "@pcmanfm . . ." from the file /etc/xdg/lxsession/Lubuntu/autostart. This is what I do all the time now.

A desktop always seemed pointless to me. You could keep a copy of the regular version of the file with the suffix -regular added to the filename and a copy of the edited version with the suffix -steam. Then just copy whichever and rename it to remove the suffix. So find your autostart file and delete the line referring to your file manager.

If you also don't want your panels just start a terminal or use alt-f2 and killall whatever-the-name-of-your-panel-is. Then use alt-f2 to start your game. If you have the disk space it might be more convenient to do this on a seperate installation on another partition. Then you wouldn't need to be renaming config files between the regular version and the steam-only version and you wouldn't need to even have panels or anything else extraneous installed. And of course no un-needed services. On the other hand if you wanted to use the same partition it probably wouldn't be too hard to write a script to rename config files and reboot in order to switch between the regular and the steam-only modes.