“Determine” is definitely the wrong word.
But yes, usually to be considered for top PhD programs, no matter where you go for college, you have to be one of the most outstanding students in your department in your class. What that means depends on the field, but it almost always includes having one or more professors really willing to go to bat for you, and then usually some form of relevant outstanding work in your field.
All that is possible from any decent department across many different colleges. However, I do think realistically, some colleges/departments are going to be able to place more people in a given year. Indeed, for some colleges/departments, they may only get a top PhD program placement every few years, and others may get several almost every year.
One obvious factor in that is just size–how many professors, how many students, and so on in that department.
But then I also think there are sometimes per capita differences as well. But then some of that is just things like where do high academic fliers with PhD ambitions choose in the first place. Like, Chicago and Swarthmore place really well in Physics PhD programs per capita, but is that because Chicago and Swarthmore take random students and turn them into Physics PhD applicants? No, it is mostly because if you are a kid seriously interested in a possible Physics PhD, and you are on track to be that sort of applicant, and you can get admitted to and can afford Chicago or Swarthmore, you are relatively likely to choose Chicago or Swarthmore! Whereas other sorts of kids have zero interest in Chicago or Swarthmore.
OK, but then even controlling for that, is it maybe a bit easier at some colleges to get the support you need for top PhD admissions? And I think the answer is . . . maybe?
If so, I do think it would largely be a function of being able to develop those necessary professor relationships, and I think that would help explain why so many small independent colleges with low student to professor ratios (aka LACs) do so well in per capita PhD studies, followed by no-more-than-medium-sized private research universities with similarly low student to professor ratios. Then the large public research universities can do OK in gross terms, but rarely compete in per capita terms, and I think that makes sense when you think about competition to really develop those necessary professor relationships. So, like, in terms of Physics PhD placement, Cal is way ahead of Chicago gross, even more so Swarthmore, but Chicago is way ahead of Cal per capita, and then Swarthmore is ahead of even Chicago. That sort of thing.
OK, so are you doomed if you go to a large public? No, of course not, many people will go from there to top PhD programs! I just think you have to be aware of what the competition will look like, and be ready to do what it takes to get where you need. And ready for Plan B if it doesn’t work out (but that is true everywhere anyway).
So to me, it sounds like you know the deal. Not least at Virginia Tech, you’ll have to be prepared to hustle from the start to maximize your chances. And it may not work out, or indeed you may change your plans, which is fine (I could tell you a whole story about what eventually persuaded me not to keep pursuing a Physics PhD, but suffice it to say I considered that a VERY rational choice). But if you start with the right mindset at Virginia Tech, it definitely MIGHT work out, so give it a shot.
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