I'm not an expert on this at all, but I had luck applying to some biophysics and computational biology programs this season (4/4 with Princeton/Yale/UChicago/UWashington). I'm writing this because I don't see you mention the GRE in your post, but you're asking for advice on how to be competitive. There are a bunch of books on the subject, but I recommend skimming "Inside Graduate Admissions" by Julie Posselt. The author is a sociologist that shadowed graduate acceptance committees trying to figure out what they're looking at when accepting students (an interesting subject given how all the students are basically qualified). Long story short, by the author's estimation, the GRE matters the most, followed closely by having some kind of background/cultural similarities with high-ranking people on the committee (but you can't control for this second factor).
One example she uses to support her point about the GRE trumping all else: the committees tend to rank applicants in Excel by GRE score before going down the list to fill their acceptance list. Further, small difference in GRE scores were, when debating the relative merits of different applicants, used as "tie-breakers." So while publications obviously help, as do good grades, LORs, etc., the professors on the acceptance committee are (I believe, after reading the book) leaning very heavily on GRE scores.
If you give it another go next year, I would try to invest a ton of time in doing well on the GRE (doing every Magoosh math problem, or taking an in person course, etc).
Some books that I thought helped a lot (in order of their relevance):
"Inside Graduate Admissions: Merit, Diversity, and Faculty Gatekeeping"
"Graduate Admissions Essays, Fourth Edition: Write Your Way into the Graduate School of Your Choice"
"Getting What You Came For: The Smart Student's Guide to Earning an M.A. or a Ph.D."