Very interesting question. Before I looked at MSAR, I guessed that 85-90% of matriculants had research experience.
A random sampling of MSAR for the first year class tells us they have the following %iles for "research/lab experience":
Harvard: 98
Stanford: 97
Wash U: 96
Case: 96
Hofstra:92
Rush: 92
Drexel: 91
Wake: 89
U MN: 89
Loyola: 89
SLU: 89
Geisinger: 88
Albany: 86
MCG: 86
Morehouse: 81
Howard: 79
U AR: 75
U ND: 73
Mercer: 73
U MO-KS: 73
U KS: 62%
So your advisor is wrong, but an interest pattern emerges. I perceive the mission-based schools place the least amount of emphasis. The research powerhouses clearly like it. But even Drexel/Albany class schools like it, or what is probably more accurate, pre-meds do research because they think it's important, but not the schools. This may also reflect the research opportunities that are at feeder schools.
I surmise that the more rural the school, the less research the matriculants will have. And sure enough:
U South AL: 83
SIU: 77
NEOMED: 30!