It can matter very much. You have a 98.5 percentile score. You have a much higher score, and likely overall aptitude & potential, than most people who will major in Mech. E. in your class.
Do you just want to sit in a cubicle and be a cog in the wheel of major corporations for 30 years? If that will give you happiness, then where you get your bachelor’s degree matters very little, as long as you attend an ABET accredited school.
Perhaps you may want to pursue a highly ranked MBA in the future, to be on a C-level path. Perhaps you will want to pursue a career in Intellectual Property/Patent law and would benefit from attending a highly ranked law school. Perhaps you have an interest in a Wall Street career as a Quant, or FinTech or IB. For these paths (and others), where you get your bachelor’s degree can matter very much.
However, the key factor is affordability. Daydreaming about schools you can’t afford does more harm than good.