A reason that LAC graduates are more likely to do PhDs may likely be the fact that, as students, they interact much more with professors than students in large universities. Moreover, those professors are going to put more effort into the interactions, have more time, and get to know students.
In large universities, the main interactions that students have are with graduate student TAs, or adjuncts teaching service courses. a large part of the courses taught by actual faculty are taught by tenure track faculty, who are more focused on obtaining grants than on being the best teachers possible, since, in every research university, and especially in the most elite ones, 90% of the tenure decision is based on the amount of money brought in with grants and the number of publications that result from these grants. Teaching just needs to be Good Enough. Teniured faculty care even less about teaching, since there recognition and fame is tied with their publications and grants, whereas the university care less even less about their teaching than about the teaching quality of TT faculty
In selective LACs, the major part of the interactions are with tenure track and tenured faculty, with a number of visiting faculty. In LACs, tenure is decided mostly based on teaching or research that involves undergraduates, so being the best possible teacher is the aim of faculty. Aside from that, people don’t go to LACs unless they look at teaching as their main career focus, so the high investment in teaching is true for tenured faculty as well.
What this means is that students at LACs are much more likely to have a positive view of people who have PhDs, as well as a close relationship with a number of people with PhDs, and are much more likely to want to emulate their favorite professors. Students at research universities will have a lot less contact, and their contact is much less likely to be personal and warm.
TT = Tenure Track, i.e., do not yet have tenure but are working towards that goal.