The solution was to place the base color in the middle of a spectrum, rather than having it fall at one end of the range of tints.
Creating a more useful palette
To resolve this issue, I wrote a script that generated a higher-contrast palette. A darker shade of the base color was placed at one end of a spectrum, the base was centered in the middle, and a 10% tint anchored the other end.
The script then generated hex codes for all the interstitial colors. Plugging them into the visualization yielded a heat-map in which patterns were easy to discern (and was also more aesthetically pleasing than the drab original).