Your understanding is…imperfect.
The Peace Corps. does not take ‘anyone, with any degree’: you apply - and can be rejected if your CV is not competitive. You will want to figure out what sort of role(s) you want and make sure that you have relevant expertise. (fwiw, the current crop of Peace Corps volunteers are very bitter about the way that they were brought home & their contracts cancelled for Covid).
Yes, students from selective schools do well in the selective grad school fellowships- but so do students from less selective schools (eg, Sewanee/University of the South) and the state universities of Mississippi, Oklahoma, Virginia, Washington, North Carolina, West Virginia, Texas, Michigan and Kansas and Montana. You underestimate the universities, the professors, and the way that schools will support their star students.
The things that will matter most in getting the fellowships and jobs you want are good grades, great LoRs and -especially- the internships you get. These internships are competitive, and essentially unpaid. That means that $$ matters a *lot: if you have to save money to pay for college or to service debt, your ability to take on those no/low paid internships/jobs will be seriously jeopardized. If the choice is Fancier Name but $100K in debt or State U and no debt, your career options will be better at State U. The best connections in the world won’t get around the fact that IR/Public Policy type careers are terribly paid until after grad school- which you typically start 2-4 years after undergrad.
*for example a recent post for a “Global Development and Public Policy Internship” looking for an undergraduate student required 32-40 hours/week of work (during term time!), with a $1000 stipend for the semester- basically bus fare