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.
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.
If you fund a UBI program by, say, eliminating preferential rates for capital income, and adding supplemental brackets above the current maximum brackets, and tie the level of the UBI to (because the former actually simplifies reporting in a way which prevents doing this exactly rather than by estimates) an estimate of the revenue resulting from those changes, there's no actual disincentive to work compared to the status quo ante due to the UBI except for very high income workers(this isn't no additional disincentive compared to those in other welfare programs from the UBI, its no disincentive at all.)
There's a disincentive compared to the status quo ante to generating income from long-term capital gains, but the practical effect of that is probably only in the upper end of the distribution (in the lower and middle ranges, the relative comparative disincentive to earning that form of income is probably more than offset by the having more surplus income to invest.)
Typical proposed implementations just increase taxes at higher income levels. So there is a taper starting somewhere above the basic income where people that are working are benefiting less than people who aren't working, and there is some income level where the basic income is effectively a pure cost. But all those people have a higher effective income than they would have from the BI.
And it likely would represent a massive increase in spending for most countries, even if it was more efficient than welfare programs. Especially if you make payments outside the tax code, which ends up with a lot of spending that is just accounting.
I assume it is funded through taxes and that people that earn more income will pay higher taxes. So once you take BI received - taxes paid, people that earn more outside income get less from the BI program than those that don't. And we are back in the same situation where a welfare program creates disincentives for work.