Global rankings are based on research output, that’s why flagship state Us are ranked higher. Sure, you can question the methodology, then there are many different rankings based on different methodologies. You can take whatever you like.
If you want to talk about teaching, it is a different story. It is not part of the global rankings though.
By the way, top national private universities do not really emphasize on teaching even thought they might say otherwise. I think they are adequate. The incentives for faculty members are all on research. Without research, there is no tenure, no salary increase, no pay, why do you think they emphasize on teaching if the incentive structure is not there? They hire adjuncts to do the teaching though.
Liberal arts schools do not pay their professors as well. They do not emphasize on faculty research, well they can’t afford it. If someone is good at research, most likely the person will be gone. They have far fewer students as compared with flagship state Us.
Liberal arts schools do not have enough faculty members who cover different fields. In 1950s, 60s, you could hire people who work in many areas within a field. Not anymore! Now in order to build a department, you need more people doing all kinds of things as fields get more specialized. If you just invest in some areas, you could miss the big market trend. So advantages to the big state U and big private universities, and disadvantages to liberal arts colleges and small private universities. Do you think it would affect undergraduate teaching and job market perspective? I believe so.
For me, school ranking depends on kids fit. If the kid knows what they want and has drive, without considering for tuition, I would take the top large private universities as they also have tons of faculty members, then next would be the top flagship state Us. I would also pay attention to rankings for the major. For engineering schools, I would probably avoid many private schools as they are really not that great.
As for several liberal arts schools that “punch above their weight” in producing PhD’s and MD’s, it is more likely that these kids are academically excellent to start with. No matter where they go, they will go for Ph.Ds or MDs.