In between Automated Insights and my current startup Spiffy, I built Teaching Startup with $10,000 out of my own pocket, 60 hours a week of sweat equity [...]
Past tense implies that you did it (and it's awesome that you did)...
More to the point, if "$250k is arbitrary because any number would be arbitrary" then it's not a good minimum baseline to shoot for, it's just a random number. Better to actually think through how the business will work, what resources you can draw upon, what your expenses will be, how your revenues will come in, etc. And then estimate your financing requirements based on the specifics of the business.