If you wish to tax them out of moral need, you hold a normative belief which is unlikely to falter, so I will not challenge it.
There is no financial reason why a 'progressive' income tax is better than a flat tax, as the 'progressive' system encourages evasion, and makes it easier, as well as greatly increasing the scope of government, requiring regulation of marriage and more.
Economically, the 'progressive' income tax is completely unjustifiable, as the wealthy invest disproportionately in economic (and job) growth, while lower income people spend more on consumption goods.
Your idea of borrowing money by way of government securities is not a long term solution, because interest gradually becomes an increasingly large part of the budget. In the long term, interest will either gradually take over the entire government budget, or an interest rate increase will force the government to default; in either case, borrowing to finance regular spending will eventually fail as a policy.
The 'inflation argument' against welfare policies is flawed as you point out, because it has never been demonstrated that wage increases cause inflation (so called "demand-pull inflation"). Whether or not some type of government welfare program should exist is an interesting question, and your answer seems to presuppose that the counter-factual is that there would be no social assistance program at all, whereas I would think that private charities would do a better job than the government, while avoiding coercion. If you require a government welfare program, I agree with you that a negative income tax is probably the least harmful, as it does not encourage unemployment.