I followed the steps on the linked page below to add the repository.
I had the exact same symptoms as the OP.
Steam still not showing in Software Center though.
I am now logged in to Steam and downloading Half-Life, which I purchased Years ago.
Nice to see some continuity in account purchases!
Instructions to add the Steam PPA
Since Steam is no longer in the Ubuntu repositories it will never show up in Software Center. I find it preferable to have the Software Update maintain the local packages in as current a state as possible. This is why I linked the page above. Yes it takes a few terminal commands.
Our packages are signed by "Valve Software LLC " You can add this key to your package manager with:
sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys B05498B7
Setup repository with:
sudo sh -c 'echo "deb http://repo.steampowered.com/steam/ precise steam" >> /etc/apt/sources.list.d/steam.list'
sudo apt-get install steam
The first command allows your system to verify that the package is digitally signed by Valve. The second adds the Valve repository so apt-get can use it. The third actually installs Steam.
I didn't want to search for the proper way to manually install the .deb package and this way allows me to keep steam updated with the rest of my system.