There are many inaccuracies in this thread, at least in my experience. I have worked in research in a STEM field for a long time, both in industry and academia, and in my opinion, there’s no easy answer to which kind of institution affords a student better research opportunities. A lot depends on the intended field of study and the particular schools in play.
I agree with @Pentaprism - It’s not true that undergrad and grad students “compete” for the same research opportunities at most universities. There are options for both, but they are separate and focused on different outcomes…
Currently I’m mentoring an sophomore in my lab (at a large public university) who has never done research before, so I am teaching him all the basics of lab technique and safety, in a very intensive one-on-one situation. I am able to do this because my lab has the personnel and the money to do so (I wish someone had done this for me when I was an undergrad…at my LAC.) Meanwhile, we have other undergrads and grad students doing more advanced research in the same lab.
Also not true - it depends on the school. At the large state school in which I work, there are amazing research opportunities for undergrads in most departments.
The same is true at my large state university - I personally have mentored UROP students who were freshmen. At my university the program is tailored to freshmen and first-semester sophomores. Incoming freshmen can apply for spots in the summer before their first year. I thought it was this was everywhere!
In my narrow little scientific discipline, there ARE pros and cons to research at undergrad- versus grad-focused schools. Most of it boils down to funding. Often, the bigger names get more funding, so they may have more projects , more up-to-date equipment, and just more bandwidth with which to sponsor undergrad research. I’d say most of those “big names” teach at universities with grad students.
Also, a larger public school may offer more options in more fields – when I was an undergrad, I wanted to do research in a particular kind of chemistry, but no professors offered that kind of research opportunity. If you really want to do research on nanotechnology or protein catalysis as an undergrad but your department doesn’t have a professor who has the funding for that…you’re out of luck.
If undergrad research opportunities are important to you, then you need to do a lot of work to suss out where those opportunities are best. My perspective is probably biased towards large-university-research, but like most other things, it depends on the student, the field of study, and the school(s).