Career choices for a developer? (Engineering Manager vs Product Manager)

Context: I work as an SDE at a Tier-1 company. I’m currently based out of Bangalore. My company has offices across the globe and it is super easy for me to relocate to some other place as well as to change roles. I am an Engineering graduate with a Computer Science major from a Tier-2 University in India.

Problem: I don’t want to continue as a developer anymore and am exploring at role change options. Both People and Product management interest me and I am not able to decide which path I should take. I think I really want to try out the PM role, but I am unsure if I will be able to scale as a PM while switching companies given I don’t have an MBA.

Questions

1. Do you think I shouldn’t care about future possibilities and just go with the Product Management path and figure things out on the go?
2. Have seen people undergoing such transitions facing problems while switching companies as recruitment processes may apply filters based on your degrees/university?
3. Any other thing you want to say.

Thanks in Advance.