maybe https://github.com/ValveSoftware/steam-for-linux/issues/3273

I've had this problem with all the source-based games I play (CSS, TF2, L4D2) and "The Plan" since, at least, the release tagged as "Built: Apr 23 2014, at 12:47:55 ; Steam API: v014". The only other game I play is Counter-Strike 1.6 and this one launches well.

I waited for an other release of the steam client to see if it fixed it but the yesterday released did not.

Since there is no popup error message when trying to launch a game (it simply freezes and won't start up), here are the console log (hoping it will help) :
Game update: AppID 240 "Counter-Strike: Source", ProcID 4872, IP 0.0.0.0:0
ERROR: ld.so: object '/home/user/.steam/ubuntu12_32/gameoverlayrenderer.so' from LD_PRELOAD cannot be preloaded: ignored.
ERROR: ld.so: object '/home/user/.steam/ubuntu12_32/gameoverlayrenderer.so' from LD_PRELOAD cannot be preloaded: ignored.
pid 4875 != 4874, skipping destruction (fork without exec?)
ERROR: ld.so: object '/home/user/.steam/ubuntu12_32/gameoverlayrenderer.so' from LD_PRELOAD cannot be preloaded: ignored.
ERROR: ld.so: object '/home/user/.steam/ubuntu12_64/gameoverlayrenderer.so' from LD_PRELOAD cannot be preloaded: ignored.
SDL video target is 'x11'
SDL video target is 'x11'
This system supports the OpenGL extension GL_EXT_framebuffer_object.
This system supports the OpenGL extension GL_EXT_framebuffer_blit.
This system supports the OpenGL extension GL_EXT_framebuffer_multisample.
This system DOES NOT support the OpenGL extension GL_APPLE_fence.
This system supports the OpenGL extension GL_NV_fence.
This system supports the OpenGL extension GL_ARB_sync.
This system supports the OpenGL extension GL_EXT_draw_buffers2.
This system supports the OpenGL extension GL_EXT_bindable_uniform.
This system DOES NOT support the OpenGL extension GL_APPLE_flush_buffer_range.
This system supports the OpenGL extension GL_ARB_map_buffer_range.
This system supports the OpenGL extension GL_ARB_vertex_buffer_object.
This system supports the OpenGL extension GL_ARB_occlusion_query.
This system DOES NOT support the OpenGL extension GL_APPLE_texture_range.
This system DOES NOT support the OpenGL extension GL_APPLE_client_storage.
This system DOES NOT support the OpenGL extension GL_ARB_uniform_buffer.
This system supports the OpenGL extension GL_ARB_vertex_array_bgra.
This system supports the OpenGL extension GL_EXT_vertex_array_bgra.
This system supports the OpenGL extension GL_ARB_framebuffer_object.
This system DOES NOT support the OpenGL extension GL_GREMEDY_string_marker.
This system supports the OpenGL extension GL_ARB_debug_output.
This system supports the OpenGL extension GL_EXT_direct_state_access.
This system DOES NOT support the OpenGL extension GL_NV_bindless_texture.
This system DOES NOT support the OpenGL extension GL_AMD_pinned_memory.
This system supports the OpenGL extension GL_EXT_framebuffer_multisample_blit_scaled.
This system supports the OpenGL extension GL_EXT_texture_sRGB_decode.
This system supports the OpenGL extension GL_NVX_gpu_memory_info.
This system DOES NOT support the OpenGL extension GL_ATI_meminfo.
This system supports the OpenGL extension GL_EXT_texture_compression_s3tc.
This system supports the OpenGL extension GL_EXT_texture_compression_dxt1.
This system DOES NOT support the OpenGL extension GL_ANGLE_texture_compression_dxt3.
This system DOES NOT support the OpenGL extension GL_ANGLE_texture_compression_dxt5.
This system supports the OpenGL extension GLX_EXT_swap_control_tear.
GL_NV_bindless_texture: DISABLED
GL_AMD_pinned_memory: DISABLED
GL_EXT_texture_sRGB_decode: AVAILABLE
Installing breakpad exception handler for appid(gameoverlayui)/version(20140520183121_client)
Installing breakpad exception handler for appid(gameoverlayui)/version(1.0_client)
Using breakpad crash handler
Setting breakpad minidump AppID = 240
Forcing breakpad minidump interfaces to load
Looking up breakpad interfaces from steamclient
Calling BreakpadMiniDumpSystemInit
Looking up breakpad interfaces from steamclient
Calling BreakpadMiniDumpSystemInit
Steam_SetMinidumpSteamID: Caching Steam ID: 76561197960340760 [API loaded yes]
Steam_SetMinidumpSteamID: Setting Steam ID: 76561197960340760
Installing breakpad exception handler for appid(gameoverlayui)/version(1.0_client)
Fontconfig error: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 70: non-double matrix element
Fontconfig error: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 70: non-double matrix element
Fontconfig warning: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 78: saw unknown, expected number
[0529/085539:WARNING:proxy_service.cc(958)] PAC support disabled because there is no system implementation
Installing breakpad exception handler for appid(gameoverlayui)/version(1.0_client)
Did not detect any valid joysticks.
GL_NVX_gpu_memory_info: AVAILABLE
GL_ATI_meminfo: UNAVAILABLE
GL_NVX_gpu_memory_info: Total Dedicated: 262144, Total Avail: 262144, Current Avail: 124104
GL_MAX_SAMPLES_EXT: 16
CShaderDeviceMgrBase::GetRecommendedConfigurationInfo: CPU speed: 1800 MHz, Processor: GenuineIntel
GlobalMemoryStatus: 2106589184
CShaderDeviceMgrBase::GetRecommendedConfigurationInfo: CPU speed: 1800 MHz, Processor: GenuineIntel
GlobalMemoryStatus: 2106589184
[0529/085541:WARNING:proxy_service.cc(958)] PAC support disabled because there is no system implementation
IDirect3DDevice9::Create: BackBufWidth: 1280, BackBufHeight: 800, D3DFMT: 3, BackBufCount: 1, MultisampleType: 0, MultisampleQuality: 0
GL sampler object usage: ENABLED
GL prefer MapBufferRange: NO
ERROR: ld.so: object '/home/user/.steam/ubuntu12_32/gameoverlayrenderer.so' from LD_PRELOAD cannot be preloaded: ignored.
ERROR: ld.so: object '/home/user/.steam/ubuntu12_32/gameoverlayrenderer.so' from LD_PRELOAD cannot be preloaded: ignored.
Game removed: AppID 240 "Counter-Strike: Source", ProcID 4877

The "Game removed: AppID 240 "Counter-Strike: Source", ProcID 4877" occured by killing the game processes manually.

I know Debian (and therefore LMDE) is not officially supported but since it worked quite well before it is too bad having the compatibility broken...

Hello @kisak-valve I have changed my PC since then and haven't tryied valve games on the new one.
But I still had the problem in october 2017.