Is the approach I take above something that could allay hesitations or suspicions on the interviewer's part about my long-term potential for their organization? Is it an effective answer to the questions raised about short term stays?
It's hard to say.
Some companies and interviewers tend to be optimistic and might feel that they match all of your criteria, and thus would have a chance to earn your employment for a long term. Other companies would conclude that you have a laundry list of job requirements that wouldn't stay satisfied for long.
But one question that will surely be asked - If you are very clear about what would get you to stick around, why haven't you picked such a company in your past four attempts?
Four short term jobs in a row can come across as a pattern of someone who will never be satisfied for long.
I understand your points about financial instability and toxic and dysfunctional work culture. But your points about expanding your skillset, taking on a major project, and more prestigious organization - those don't seem to match any of your "what it would take for me to stay" requirements at all.
You need to be able to answer "Why will this time be any different than the past four?" I'm not sure your script does that.