There are two basic ways to provide funding, either you force people to ask for it and deal with a long approval process, or you trust them with it in the beginning and subject them to an audit process. Approval processes suck because they take too long and the bureaucracy stunts your ability to react to changing conditions, which is why most private sector companies use audit processes internally.

And even though the public sector incurs real costs due to an approval process (you're not just paying the salaries of the overview bureaucrats, but also the salaries of the project workers who need to sit around and wait for funding to come in), it persists because it is viewed as more legitimate and more protective of legitimacy.

If you ask the public whether they would prefer for a public program to cost $100 million, including $20 million which ended up being embezzled into private pockets, or $150 million spent entirely legitimately, most members of the public would rather spend the $150 million for a legitimate program, even if you could show them how the additional taxes needed directly affected their personal bottom lines.

The reason why we don't give poor people cash directly is simple: even if 99%+ of beneficiaries will use the money responsibly, a tiny minority will not. That tiny minority will get written up in the press, and the perceived unfairness will politically doom the program. How public money is spent is, unfortunately, important.