Such a system looks a lot like the systems which are currently used in the Netherlands and Scandinavia. True, there are some differences: BI is called 'unemployment benefit' for those who don't work. And for those with a low income, there is no BI, but instead there are subsidies on housing, health insurance etc.
The personal standard deduction is not "income tax"βit's the opposite of a tax. It's the amount of your income which is not taxed.
This system wouldn't even necessarily require progressive taxation (though I would favor it). A flat tax rate of even 40% could be more than enough to provide for BI and everything else (and everyone would receive their BI tax-free).
It's very different from unemployment insurance because unemployment disappears once you start working. So there's no incentive to take on jobs which pay less than unemployment, or even to take on jobs which pay less than (unemployment + opportunity cost of working).
It might be helpful to do some research on how taxes (particularly deductions and marginal rates) actually work.
So your first $10k (from BI) would be tax free. If you don't work, you don't pay any taxes.
Let's say you have a part-time job which pays $20k a year and the lowest marginal tax rate is 30%. You'd then pay $6k in taxes on that $20k.
Thus, your overall income would be $24k, much more than you got from just BI. This works if your only work is a single $100 lawn mowing job. BI, when coupled with an adequate standard deduction, eliminates disincentives to work.