Good luck with your search.

Free advice from the Internet- That role you're describing is pretty rare for new grads. You'd normally look for someone with experience and a track record before trusting them with open ended work or product management roles.

Start by being a "junior" builder in a team, then as you prove yourself you'll be given broader scope, this can take a while. There are teams building things that need strong builders. The smaller the company the more likely you'll be able to grow faster if you perform well.