Others have already noted that most MD programs that you could apply for will have research opportunities, but if you want significant exposure to research, you should choose from the curricula about how much "research" is encouraged in the form of independent theses. Sure, many brand schools involve some formal student research office, and you should look for them. But they exist at a lot of medical schools, and not just the brand-names.
Unless you have dedicated time in research like the MD/PhD's, Mayo, or CCLCM (off the top of my head), don't think you'll be involved with making your own new strain of gene-modified mouse to study disease. Be reasonable with how much research you want to do while keeping your head above water with your medical school obligations. (Really, ask the folks in the Physician Scientists forum and APSA.)
Others in the meded community can comment on the residency match rates correlating with brand-name schools, but your description I think oversimplifies (incorrectly) the true complexity of residency matching.