I noticed it's not very straightforward where the save games for Tomb Raider (2013) are stored. Depending on how you purchased the game, it's in a different location.
Note that it is not really straightforward to restore the saves afterwards (see below). For that reason, it's better to enable Steam Cloud and have your files synced to the cloud.
Note that the ID for the game may be different, although, if you have the latest version, it should be as displayed. In any case, if you can't find it, try searching for files called profile.dat or save1.dat.
A slow and unresponsive Steam app on Mac is a rather common issue. Symptoms are intermittent display freezing of the app, sluggish and delayed reaction to user input (i.e. clicking on items) and slow animations in the various screens. The issues may not appear straight away after starting Steam but may build up over time, causing Steam to run increasingly slower. However, the problems are only related to the Steam app, as games themselves run normally.
This is a known (but undocumented) issue with GPU acceleration, which luckily can be turned off rather easily:
With the Steam app active, in the main menu, click Steam → Preferences.