"you're competing with other resumes"
Very true. One of the best ways to not be compared is to approach things in a way that there is no competition. Approach employers when they are not advertising. I did this once and got a job that in no way I was qualified for resume wise. Once a job is advertised or with a headhunter you're being compared to someone else. (Same with dating most of us would rather not have to compete head to head with 20 other people and hope to be chosen the best of the lot.)
Just curious to know, I am normal person or a mere mortal as you described.
How do I get hired without the normal Resume, Recruiter, HR, Interview process?
In my experience, anything that gets around the formal hiring process is a win. Finding a way around the application process is the application process.