Honestly, one thing that changed my opinion about rankings was that I once did an exhaustive LinkedIn search to see which MFE programs placed more students in firms like HRT, Optiver, Jane Street, and the likes. And I discovered that Boston University (at the very bottom of the ranking list) placed some alumni in Jane Street or other top prop trading firms more than I could find for programs like Georgia, UCLA, and even NYU. I understand some don't update their LinkedIn profiles but if Boston MSMFT students can get into Jane Street (meaning they were considered over candidates with Bachelor and Masters degree from the "top/prestigious" schools, think Harvard, Princeton, MIT, etc.), then it's really not about the rankings of the schools you attend.
That being said, if you gain admission into the very top schools like Princeton, CMU, UCB, Baruch, it is sort of easier to get interviews because of the extensive alumni connection. But if you are attending a lower ranked school, you will have to make up for this by networking, connecting with recruiters and tirelessly attending their career fairs. Get information about which schools they are planning to recruit from (could be the business school or the finance/FE program itself) and find your way there to talk to a recruiter. Then, submit your CV and hope for an interview. During interviews, prepare hard and put in your best. Connect with colleagues in other "top" programs to see how they prep for interviews. You do this again and again until you land a job.
The real competition is not during the admission phase itself but in the job market. Goodluck to us all!