Most universities in the US have a very complex mission. Research is always "valued" but not always required. There are many primarily teaching colleges and universities, but even those put some value on research, though it is normally less than that on teaching. The charter is normally threefold: Teaching, Research, Service. The emphasis at a SLAC (Small Liberal Arts College) might be 60/30/10 or similar.
At a large R1 or R2 university, however, the first two numbers are reversed (approximately): 30/60/10. But it isn't true that research focused universities don't do a lot of teaching, even undergraduate teaching. There are a few research-only places, but not many. They may teach doctoral students.
Here the numbers indicate the expected amount of (successful) effort devoted to each area for career advancement. A 60 implies a lot of effort - the primary focus - with some evidence of success for that effort.
The "value" however is a bit harder to pin down. The value of research might be (hopefully) the value put on advancement of knowledge. But it might also entail an increase in the perceived reputation of the institution. In reality, it will be a combination of such things. There is (informal) competition between universities for things like favorable mentions in the press or for total grant funding or Nobel Prize winners or ...
Many of the state supported R1 institutions have 40,000 or more students, the vast majority of which are undergraduates. Their courses are normally taught by professors assisted by (many) teaching assistants. Doctoral students are funded with TA positions and a department that teaches a lot of undergraduates (Math, English, ...) needs a lot of TAs.
There are also a relatively small number of research only institutions, many funded by the federal government. They might have in-house researchers and they might give grants to researchers, though in that case the researchers are probably university professors.
Many corporations also do some important research, though most industrial research is product based rather than pure research.
But most research in the US is at universities.
At a SLAC, a professor might teach around 4 courses at a time and, since the institution is small, those courses won't have a lot of students; perhaps 30. Students have a lot of potential for interactions with professors.
At an R1 institution a professor might teach 1 or 2 courses at a time (I've had both experiences) and they might be large or small. Upper level courses are likely small (30-40 students), but entry level courses might be huge (few hundred students). The professor lectures a few hours per week and the students meet with TAs in small groups (30, say) a few hours per week. There is little opportunity for interaction with the professor.
At either sort of institution it is possible to arrange a term/semester in which there is no teaching so that one may focus on research. This is more likely at R1 institutions and fairly likely to be grant funded - the professor "buys out" of a course or two with grant funds.
So, it is incorrect to say that the various institutions "don't care" about either teaching or research. All do. But the balance and emphasis is different.
Note that many R1 (extensive research) institution in the US are funded by the states (as well as by tuition and grants) and the states fund the institutions out of a desire/need for an educated citizenry, though, it recent times it is largely to support industry's need for educated employees. So, even the research institutions have an important undergraduate education mission along with research and graduate education.
There are also a small number of private R1 universities (e.g. Dartmouth, Harvard, Carnegie-Mellon...), though these tend to be very old and many of these were once SLACs.
I personally studied math at a SLAC as an undergraduate and at R1 universities for graduate education. This is fairly typical, though many others did undergraduate studies at R1 places, though the actual undergraduate education is very similar. My undergrad institution has been putting more and more emphasis on research over the many years since and now has a much better research profile.