Chill. Hiring is broken. I am Product leader at a Unicorn. From a tier 3 college. Non-mba. Startup experience.
How to navigate: Irrespective of what you have studied, what you have done so far, you can easily get into PM-ing via referrals up to SPM level. In your situation, ONLY referrals would work. Job boards work if you have logos (IIT etc, or say OLA etc). Because HRs have a shortlisting criteria. Also, bottom of the pyramid, there is just too much supply.
After SPM level, it gets super tough if you haven't worked in larger product orgs etc. At GPM/ AD & above, folks want to only hire if your profile has enough experience working in some known product orgs.