Hey there - so there's no right answer here - it's a pretty subjective thing. You're right that you'll be pulling double duty when you visit but you're also, to a certain extent, overestimating how much of the visit you'll be able to leverage in the interview. The reason I say that is because you'll need to bring specific things about the school that tie to you and the visit could be, and often is, general.
As for flying across the country - yes, no school will view that as a pure check the box exercise but that doesn't change the fact that its utility may be relatively limited. Visiting before you apply is less "check the box" to be honest because it's immediately useful. You can talk about it and the school would think a reasonable adult who has visited is applying because they like what they saw. Between application deadlines and interview invites, what would your reason be? If the adcom asked you as you walked in, "why are you here today, and not 3 weeks ago?" what would you say? Understand that relatively limited does not mean useless. But if you visit now - fly across the country - and then you go back 2 weeks later for an interview, what'll be the point in that? Because your question was will it help with getting an interview? And the answer is, at this stage, it's difficult to say. By the time you land there, they might have already reviewed and closed your file. Or maybe they won't have. Or maybe they'll be reading it RIGHT THEN.
These forums are great because you can get a lot of opinions on things that really matter to you right now but there is no "right" answer to most questions. So what I will say is this - if one of our clients asked us this, we would tell them to not. I hope this helps and serves as another data point to consider. Is it the holy word? Of course not! It's just my professional opinion.
Good luck and enjoy your visit - whenever that might be

Bhavik