The easy way to fix that gap: start your own business. It doesn't have to be successful, but it fills the gap.
I like to also make merch and take video interviews while wearing say a branded shirt to show I was serious (I was not, it cost $20). I’m also not afraid to offer a screen share and walk through infrastructure diagrams or other intriguing IP to show something I have been working on instead of publishing everything to GitHub (unless it’s an OSS project)
Mind you this has backfired on me. I took an interview with a series A startup and the ceo flat out rejected me upon hearing my startup. He wished me best luck but stated he needed someone who was 100% invested in his startup and not someone thinking about their side hustle. Couldn’t fault him on that and instead flipped the interview into a Q&A with a billionaire. YMMV
Your idea is interesting but might be more elaborate than you even need.
What's weird is that most employers reject the application after going through all the interview phases successfully, without mentioning any reason, but it's pretty obvious that once the "upper man" has to give the ok, they are not fine with someone who has/had their own side-projects.
I did start my own company, originally to honestly make my own mobile games. A spent a year experimenting with all sorts of stuff, and then just started freelancing for others, and was a lot more successful at that. I probably still have some half-finished attempts at mobile games somewhere.
Whenever I try going back in the job market, I somehow seem to be "over-qualified". They don't want someone who can do anything needed or who is able to manage on their own, they want someone who does one thing and well. They want someone that is most likely to "need" the job forever, without posing any danger to the leadership positions.