The most important factor is research in the area of interest of a given student. Whether or not the university as a whole has programs in other fields is completely irrelevant. A prospective student should evaluate a given department based on their own research goals. If the department of choice is at a STEM-focused university, then great. If it is at a more broad university, also great.
It just so happens that many of the public, land grant universities that, as a whole, have very broad undergraduate offerings also have some of the best engineering programs in the US. In addition to the ones that @eyemgh pointed out, narrowing the focus to a STEM-focused university also eliminates other such schools as Michigan, Purdue, Virginia Tech, Texas, Texas A&M, Florida, Penn State, Colorado, Washington, Maryland, and many others that are broadly-focused universities that also have very significant engineering programs. There really is no reason to artificially reduce the scope of one’s school search like that.