Thanks for all the positivity! The interview process was really so professional and engaging, so I would assume they're really sticking to some kind of process for admissions. I didn't really get the vibe that reaching out to them following the deadline would be of any benefit, which I think is one of the reasons that they told applicants about the March 15 decision date. I only interacted with faculty in open-to-all-comers situations; in this case, just the open house and some open colloquia. I do know someone in the program who talked to the faculty about me prior, so they had some introduction to my work, and thus those conversations maybe had a little more substance - but, I never had any sort of one-on-one meetings outside of scheduled events.
As long as you haven't gotten a rejection notice, I would assume you're still in the game! From what I understand, they got far more applicants this year than in any prior year, which just takes that much more time to go through them all. I hope you get some clarity and good news soon!