If basic income is X, some of the individuals receiving it will have a tax liability >= X. All legitimate UBI proposals I've ever heard share this property. It's either that, or the government finances the entire UBI with a budget deficit. And no one is seriously proposing that.
There's also minimum guaranteed income... which guarantees that Earned Income + Basic Income - Tax Liability >= Minimum Guaranteed Income, for all individuals. That's a stronger proposal. We should really aim for this.
But I don't think implementing UBI means that everyone is better off the next day. Some people's post-tax income will stay the same or go down.
Another question: do UBI and income tax go together? It would be interesting to see proposals that provide UBI but eliminate income tax, replacing it with some other kind of tax.
As a thought experiment, imagine that you increased the UBI payments so that they took up all the gross income of the country. Now your tax rates are 100%, even though the "net impact" of the redistribution is 0. A smaller UBI is the same concept, just the percent of income distributed is smaller. The bigger the UBI, the larger your tax rates on all workers. As those tax rates reduce the supply of labour, the costs mount even more.
A $20/week UBI for all of them would cost $2000/week. You would need a flat tax of 39.6% on everyone to cover that. So everyone's marginal tax rate increases, with the associated deadweight losses.
tldr; you can't just hand wave and say a redistributional tax scheme is free since it all balances out across the population.
But interestingly: basic income provides more benefits with as income inequality goes up. That is, if everyone in society already has the same level of income, there would be no way to finance basic income through income tax. You would collect the same amount of tax from every person and then give it right back to them. However, in your hypothetical distribution, that one person at the top is singlehandedly providing the basic income of 2% of the population. And we are using your proposed very fair taxation scheme of flat tax. And in reality, income distributions are far more skewed than your hypothetical distribution.