@intparent thanks for calling my method “lousy.” 
I warned you it was back-of-envelope, and I tried to be balanced.
The word on the street, and probably what @firstsax is referring to, is that students can literally walked into a myriad of labs at Caltech and get research experience. I don’t think it is that far from the truth.
If you look at research funding for engineering, I see $465 mil for MIT versus $110 mil for Caltech. On the other hand, there are circa 360 Caltech engineering undergrads and 1750 MIT undergrads, if I can trust US News. Again, rough calculation, and only for a subset of the students, and I am not going to hunt it down for every major or field, but it shows they are in the same ballpark of engineering research expenditure per engineering undergrad.
With regards to Mudd, I do think Mudd IS in fact going to be quite excellent for research opportunities - probably one of the best.
But if you want to put it more on the same footing, you can start with another assumption - I read somewhere a typical PI has 6 positions. That would yield a result that is needlessly unkind to MIT, so let’s bump that up to 10, and ignore postdocs.
Then we have:
3200 spots left for 4500 undergrads at MIT
1800 spots left for 1000 undergrads at Caltech
940 spots for 800 undergrads at Mudd.
So in terms of sheer accessibility, I think Caltech is hard to beat and I agree with firstsax . I hear you on humanities students, but they do research too, and Caltech has them as well, so I don’t think that changes the picture materially.
I did already point out MIT’s Harvard advantage. (Caltech has UCLA and USC, but the distance is daunting, especially given LA traffic.) And the total research spending is amazing at MIT, so there will be amazing things going on. And finally, one more thing I put in MIT’s favor - due to the sheer number of opportunities on an absolute (versus per-student) basis, the more focused student is more likely to find exactly what they are looking for.
So just because Caltech wins in number of opportunities per student (if you agree with what I am contending), it doesn’t mean it is the best choice for every student. As demonstrated by MIT’s winning cross-admit rate. Each place absolutely has its pluses and minuses, as surely anyone without an agenda would agree.