I can speak from experience, not as the hiring manager, but as the person who got the job without the formal qualification.
I've been working at this electronics company for ten years now. I was hired into a customer service position and worked my way up into a supervisor position. I'm a high school dropout without a degree although I did get my high school diploma later. I applied for the customer service job with this company specifically for the reason of getting my foot in the door, and hoping to work my way into the position I wanted.
I'm a huge electronics nerd. I've been building electronic equipment from scratch years before I started with this company. Etching my own printed circuit boards, populating, troubleshooting, etc was what I did on the weekends. I started getting into hobby CNC and building CNC machines for fun. While I have a "hands on" skill set I don't have a degree which covers the theory side of things. That makes you an incredibly hard sell in the field of electrical engineering.
So 8 years went by working for this company until a technician job opened up. The job was for fabricating test PCBs...the thing I've been doing about home for 15 years at that point. The PCBs happened to be fabricated using CNC machines...another thing I was familiar with. The problem? They were looking for someone with an engineering degree that had a background running a PCB fabrication line. I e-mailed HR and told them that they aren't looking for someone with a degree because you don't learn any of this in any classroom, and someone running the floor of a fabrication shop doesn't have CNC experience because all PCB fabricators etch their boards...not mill them. I told them they were looking for a unicorn, and I want to apply for the job. I told them if they can't find a qualified person I'm eager to prove myself. I got an e-mail after a few weeks telling me they'd allow me to interview.
I've been at the job for two years. I've reduced the costs by a huge factor and sped up the process to an insane level (all technical terms). I don't think there is really anything you can put on a resume to convey that you are "qualified" without having qualifications. It took me 8 years working in an unrelated field and basically telling the folks doing the hiring that they were looking for a person who doesn't exist. My best advice is to find a position in a company that you are qualified for. Make sure everyone in the company knows your interests and aspirations. If you are a good employee eventually you'll get an opportunity. Or you can go to school.