That is a rough situation, and it sounds like you are hanging in there impressively. Hats off to you for that.
I do have to agree with the other poster, I sounds like the big thing here is the framing of the situation, your presentation, and projecting confidence.
Perhaps doing fake interviews with people and getting their feedback on the non-technical aspects could be helpful (it sounds like you are nailing the technical stuff). Or you could record yourself answering expected questions and work on coming across amiably and confidently, as it fits your personality of course.
Also, I'm not sure where your located, but my guess is that there may be some geographic/cultural differences in the severity of this issue. My impression is that in SF, one's current employment status is less important than other indicators.
I was living and working on the east coast in a major city for about the past 8 years, and feel that culturally I resonate more with urban lifestyle, tech & university hubs, etc. Sense of humor is important to me ... I'm one of those people who was raised on The Simpsons, despite how cliche it is and how much Sean & Hayes of Hollywood Handbook would mock me for saying so.
The family situation has required me to move to the rural midwest to do full-time care and administration of a legal issue, provide childcare, and do many other things. I went to high school and college here, so I know the ropes, but it's not a place that I identify with or culturally fit into, and I think anyone who interacted with me for a few minutes would know that quickly.
I will always continue practicing presentation skills, but confidence in public speaking and especially confidence in speaking clearly and precisely about the technical work I've done are things that come naturally to me.
Firms from SF have been among the worst. The tech culture there just seems problematic and I think I would strongly prefer New York or Boston over SF, though for a good-fitting role, I'd consider really anywhere.
But after interviewing with a lot of SF firms and getting feedback that either (a) they would not consider paying adequate relocation or (b) they would not consider someone who is not currently working, I just focused my search on different geographic regions.
I tend to get a lot of first interviews based on my education and past experience, but after I answer questions about my current unemployment gap, it's like everyone just feels it's too complicated to bother fiddling with it at all and they just dismiss it without considering my skills.
Your education and past experience are very impressive. Your employment gap is less so. There's a lot of passion and pride in your stories, use that energy to sell yourself to employers - tell them you'll make something for them that they'll be proud of.
What about giving tech presentations as self-employment? Since you enjoy doing these, you would just need an audience, maybe at a school? Or maybe presenting concepts on YouTube?
"There were problems and I had to move home to run the family for a while, but I stayed busy making YouTube videos about tech concepts," might be a more palatable narrative to a prospective boss. The kids could even help make the videos.
As I mentioned in the other comment, given the full-time duties I have to care for my current family situation, there simply is no physical possibility for doing anything that remotely resembles work. Devoting the time necessary to create presentation materials alone would be way, way too prohibitive. Already, even just spend a couple hours on telephone interviews and a couple hours on personal code projects is an extreme time strain for me, not to mention that in this rural area there is no reliable internet connection and I often go for hours, sometimes even days, without ability to connect except by tethering to a 4G device and paying huge data charges.
If I have to pick between spending those few hours actually coding vs. spending them making videos or something, I think it's obvious that actually coding (and reading to stay current) is much more important. That's what I want to do in my next job. Making YouTube videos is likely to be completely ignored by most recruiters, and even when considered it won't be regarded as useful in the way that actual coding skill will be.
I do have to agree with the other poster, I sounds like the big thing here is the framing of the situation, your presentation, and projecting confidence.
Perhaps doing fake interviews with people and getting their feedback on the non-technical aspects could be helpful (it sounds like you are nailing the technical stuff). Or you could record yourself answering expected questions and work on coming across amiably and confidently, as it fits your personality of course.
Also, I'm not sure where your located, but my guess is that there may be some geographic/cultural differences in the severity of this issue. My impression is that in SF, one's current employment status is less important than other indicators.