No, what you are saying simply is not true. Some "liberal arts colleges" have PhD programs, like Bryn Mawr College, Wesleyan University, and Tufts University. Some have nursing colleges like Adelphi University. Some "research universities" are called "college" like Dartmouth College. In the USA, the words university and college are used interchangeably, without firm rules regarding the types of degrees that they can offer. The word "school" is overloaded because it also refers to pre-college education.
Everywhere I see people talking about / posting about the differences, the tone of writing makes it seem like these are mutually exclusive sets of "schools". Is there something about the operations of a research "school" that makes it incompatible with the values of a liberal arts "school", or vice versa? Or is it just a coincidence that there isn't any such institution?
They are not mutually exclusive. Some liberal arts colleges weight research equally important as teaching. Examples include Bowdoin College, Wesleyan University, and most of the NESCAC (Williams, Amherst, etc.). Meanwhile, some "research schools" weight teaching as roughly equally important to research, especially R2 universities. Furthermore, some research universities have a designated teaching track, where professors can get tenure based solely on teaching and not at all on research. A great early example is Uri Treisman at UT Austin. Harvard University has the preceptors. I've also got a friend who is a teaching professor at UC Irvine. These kinds of jobs are becoming increasingly common because even big research universities recognize the importance of good teaching. Lastly, this classification of "research only" vs "teaching only" seems to leave out regional universities, which make up a huge percentage of tenure track jobs in the USA. I don't know much about them, but I know that success at such a place requires both teaching and research (not sure how they are weighted).
What is true is that there's a correlation between "liberal arts college" and weighting teaching the same or greater than research, and between "research university" and weighting research the same or greater than teaching. However, the values at these types of institutions are not "incompatible." Even at a big research university, the faculty and administration do not want students to fail. Even at a small liberal arts college, the faculty and administration do want faculty to publish, become respected scholars, win grants, etc. Some extremely strong researchers work at liberal arts colleges because they prefer small classes, incentives to develop deep relationships with undergraduate students, a well-resourced environment where funding does not depend on the (often crazy) state legislature, the ability to teach a wide variety of courses, summers that are truly "off" because there is no expectation to remain on campus, etc.