The profile you make has two parts, one is the LUT that gets loaded into the graphics card. Every single program managed or not, even the desktop uses this unless they toss it or use a special sort of video layer.
Almost all games toss it (although apparently there is a way to lock it). Blu-ray software appears to either toss it or use some protected video layer or something that won't make use of the LUT. HDTV and DVD stuff some players will and some won't.
The LUT is used to set the tone response curve. The sRGB standard uses sRGB TRC. Many calibration tools and monitors don't support that and only color managed apps will actually deliver you the proper sRGB TRC. AdobeRGB uses Gamma 2.2. Gamma 2.2 is pretty standard way to set the display for movies/games/native editing. Although sRGB photos and web stuff actually matches perfectly to sRGB TRC and not Gamma 2.2.
The LUT also at the same time sets the white point.
Then the second part is the profile of screen. Only fully color-managed apps make use of this. Basically just stuff like FPV, Irfanview for jpgs with a plug-in, Photoshop/After Effects/InDesign, Photomechanic,Windows Photo viewer/gallery for jpgs (and only when not in full screen or slide mode
for some bizarre reason),some OEM photo and scanner software,Firefox,etc.
Some are simple matrix based and just essentially record the positions and luminances of the primaries and secondaries. The programs can then take the matrix for the assigned profile of the image and multiply it to transform it into the monitor's colorspace. Some are that plus more complicated correction tables instead.
Anything that doesn't toss the LUT or use some weird video render mode will automatically make use of the LUT (so long as you don't forget to tell the OS to load it into the graphics card LUT).
Partially managed programs may only do simple things like transform image space into sRGB and do nothing more, they won't use the monitor's profile (beyond whatever the LUT did).
Fully managed programs take the source profile and monitor profile both into account.
Some browsers don't bother managing images unless tagged (safari) and some such as Firefox can be set to automatically assume sRGB for everything not tagged otherwise and will manage everything on the web from all the tons of untagged sRGB images to the font colors and graphics, etc.