How much do 6-stop ND filters affect color, and are X-Rite profiles more accurate than Adobe/DNG profiles?

As you can see, with proper color management, the different tints of each filter do not significantly matter. Both filters can be properly corrected to match a calibrated image with no filter. That's because your color management software is adjusting each one differently to give the same result based on what it thinks the color should be.

The most obvious differences in the sample images are that, for whatever reason, the images made using the DNG Color Profiler are much more saturated than the images made using the Adobe Standard and X-Rite Color Checker Profiles. Those two are probably more accurate renderings of the scene than the DNG Profile. It's fine if you want to boost saturation as you process raw files, but it's usually better to get as accurate a rendering as possible with the color profile applied and then increase saturation later in the process.

Also note that with with every render, I clicked "auto" in ACR. I suspect this may cause some controversy, but at best it fixes any small exposure differences.

In the far left column, whatever minor differences there are almost certainly more attributable to the differences in the scene (no filter box, filter box with mostly blue and white colors both in sunlight and in shade, filter box with red, gold, and white colors in sunlight and shade) than the differences between no filter, the B+W filter, and the Breakthrough filter. The entire contents of a scene can affect the Auto White Balance function of the Adobe Standard Color Profile, which analyzes the entire scene and attempts to set color temperature and tint based on clues in the scene.

In the middle column you've used Adobe's DNG color profiler, which should be based on a color profile you created/edited. If you didn't edit anything in the DNG Color Profile, it's still using the default color profile for your specific model camera and, again, guessing at what part of the image should be neutrally white or gray. I assume you used the color checker with the color profiler to create a custom profile? I've never used DNG files or created a color profile in Adobe (I use different raw conversion tools).

In the right column you've used the X-Rite software to sample each of the squares on the color checker and correct each image to make the colors display what the X-Rite software says they should be. If you properly created an individual profile for each image, then there should be virtually no differences in color between each of the three frames, and there aren't.