Exactly. To be honest, anyone with a desire to work in public policy at any sort of professional level is going to want to pursue a MPP (Masters of Public Policy) or MPA (Masters of Public Administration) program in graduate school. Or some equivalent course of graduate studies like a law degree or MA in something like economics.
A lot of the higher level public policy internship and fellowship programs basically hire out of grad schools not undergraduate. The most well known one for those wanting to work at the federal level is the Presidential Management Fellowship Program, which recruits out of Masters degree (or PhD and law) programs. https://www.pmf.gov/
As a former Presidential Management Fellow and graduate of a masters degree public affairs program (in my case the UW school of marine and environmental affairs) I think the best undergraduate path would be something that provides both strong writing and analytical/quantitative skills and probably some economics