Unless you are in a field where the PI hires doctoral students directly into their lab, you aren't likely to get a response from anyone prior to admission. This is especially true if you are applying with only a bachelors.
But, in any case, most universities have a requirement that you pass a set of preliminary/qualifying/comprehensive exams before you get very serious about doctoral research. This is to assure that you have a broad knowledge of your general field and may require some advanced coursework to pass them.
Since you haven't gotten a reply, then I'll assume that this is the situation at this university.
I don't think that mentioning three professors is necessarily bad, but it isn't really necessary either. If all of them are in a fairly narrow subfield that interests you, it might be enough just to name that subfield.
However, don't overly focus your SoP on these people, or spend too many words on this particular thing. You want to convey your future goals and plans both for your studies and thereafter. Some specificity is needed for this so that you don't appear to be unfocused.
The advantage of the US system, with admissions handled by committee and advisors chosen rather late, is that you get to meet potential advisors before you make a choice. There is more to consider in choosing an advisor to assure that you find someone who is both compatible and who will also provide the appropriate level of support. You might even be able to take a class with a potential advisor and establish some rapport. If you have to choose only based on their field, it is possible to make a serious mistake.
So, naming three is a bit of overkill, but not bad, as long as you use the available space in the SoP to say all that needs to be said.