I am currently a postbacc student at NYU. I highly encourage you to apply wherever you want to apply, as long as application fees are not too much of a burden financially. In my experience, postbacc students are a good way for institutions to make money without having to report on undergraduate acceptance/graduation rates, so I did not feel like it was as competitive as I thought it would be. Of course, every school is different.
I was accepted to both NYU and Columbia... having an undergrad degree from Georgia Tech and work experience in healthcare consulting and advertising technology. Nothing out of the ordinary. My other cohorts had similar acceptance. The point about postbacc is that they do not expect you have had healthcare experience. I was accepted to NYU with "clinical reservations" because I did not have that much experience in patient-facing healthcare, but that just means I have to get shadowing experience while I am in school. I don't know why I was singled out for clinical reservations because I have classmates who were professional bakers or also in advertising who did not have these reservations. I also don't think that NYU is going to kick me out anytime soon because I am making good grades and paying tuition!
Bottom line is, if this is what you want to do, definitely go for it. Your background is less relevant than you think and it sounds like you have good experience and education history. You may find you do not even need to wait a year to apply, but of course, that is up to you. I'm sure that getting a year of clinical experience would only benefit you. I recommend reaching out to the programs you are interested in (and even some you are not) and attending information sessions.
Happy to answer any other questions you may have - good luck!
