I am not really sure of the requirements of the jobs you are seeking... How much "proof" do they need? What do you need to show to show you can do it?
With my kind of work, I used "freelance" projects for my resume to show "experience" and get my foot in the door.
My first job in college was as a cashier, and they wanted retail experience for a min wage, entry level position

. I claimed to have done some work for my step-dad's business, which was true, but at the time it just felt like earning some allowance (I was not an employee...). The interviewer recognized the name of the business and made lots of positive assumptions, and I was hired! Prior to that, I had been on several min wage retail jobs and they all were skeptical of my lack of experience.
I did similar things after college when seeking design work, which usually requires a portfolio, degree
and experience.
All of my prior work experience was conveniently done for family friends or family with different last names

Turns out those people make great references too!
Of course, working in design, having a tangible portfolio helps as "proof", and it is easy & even expected to claim freelance experience.
Basically, get creative with "work experience". If you can, invent a project, hire yourself, and produce some tangible evidence of this to back-up that experience.
I agree the expectation is unreasonable, which is why I have no pangs about working the system a bit.