Help with Illustrator blending modes

Create a new document in CMYK mode.
Draw w square and fill it with #0065D7

Thius color is RGB format and OUT OF CMYK COLOR SPECTRUM.

Save the document and reopen the document.

If you are correct, the color value of that rectangle will be automatically converted to CMYK and the out of gamut data for that sqaure will be lost. In other words, your color panel CANNOT identify the color you applied because your color was OUT OF GAMUT and the claim is that the colors are being forced in CMYK Gamut. 

When you export the file or print the file, THEN A VALID CMYK COLOR PROFILE can be used to convert your colors to CMYK. Before then, the CMYK Document Color Mode does not destroy the extra data ffor out of Gamut colors because there is such a wild difference in CMYK Output profiles.

For example, you pick an output profile SWOP which is weak in the Red Orange gamut, for example. Your active document does not change your selected color to cut off the Red Orange colors. It keeps the out of gamut colors. Then, when you change to some other CMYK profile, it has MORE data to work with and can better match the intended output profile. This is the only way that it makes any sense.

So imagine you swapped back and forth between output profiles while working with your art. Eventually you could technically lose all the details in your work because you keep cutting off parts of your gamut that each different profile misses. That would produce the worst possible results.

So since Adobe HAS TO SHOW THE COLORS IN RGB ON YOUR SCREEN, why not just store the RGB data (that they have generally coallesced into the compatible CMYK subset of RGB) up until you are ready for export or print? It helps them, and it makes your prints better.

If you can show me where in Adobe's documentation they claim that using CMYK Color Mode forces adherence to CMYK Color Moel Gamut during the working process (not just during output), I will be convinced and it will be just another reason to avoid the CMYK Document Color mode. I have a list of at least  a dozen reasons to avoid it in the first place, more fodder for the list would be greatly appreciated.

I think you are mistaken here.

To test this, I create a new document in CMYK Document Color Mode. I draw a rectangle and fill it with an RGB color that is outside of the CMYK gamut. According to your logic, that should not be possible because the color will always be within the limited gamut of CMYK.

There would be no need for Gamut warnings in CMYK mode if Adobe limited your available colors to the CMYK gamut.