I am an a senior in university with (paid) research internships at top labs, and I got into a top 10 university in the US for my PhD which is essentially fully funded (including stipend). I might stutter, but when I present novel results to everyone at the lab meeting, they listen intently. Members of the lab respect me and let me finish my sentences. There are no awkward face reactions which I find talking to other people. Elite scientific research is one of the best jobs for me. Although stuttering should not hinder you in any elite field of human endeavor.