I had a fairly senior hiring manager for an investment bank call me a liar to my face, basically because he couldn't understand how a consultant can have more than one client at once, and so my CV is non-linear! I dodged a bullet there.
Initially Google used to send US cheques, in USD. So I had to go to the bank to deposit those in my personal account, which required a branch manager to sign it because 1. Who is Google and 2. Why is someone getting a large-ish cheque from a company overseas (AML laws).
Anyway, at some point I've decided to incorporate a company. Trying to explain to my accountant what my company was doing (online publishing), and trying to explain to the bank this was not a money laundering operation required some effort.
People had no idea that one could make money on the Internet.
Advertising for them had to fit in a TV, billboard, radio or cinema square. Anything else was science fiction.
I then left my 17-year full-time as a mainframe developer (yep) to run my company full-time. You can't imagine the run around I had when I tried to find a home loan for our first home a year later - how do you make money? was always the question. It didn't matter that my answer was "I am the owner/director of a company". They wanted to know what my company was doing and just "online publishing" wasn't enough. Even with tax records and everything.
So I empathise with the OP on how to demonstrate skills and experience while being self-employed.