The short version of the answer is, they didn't realize it would be that hard when they started. When the scientist-administrator Vannevar Bush pushed Roosevelt to authorize it, he thought it would cost $400 million, and they thought it would take about a year to complete. That's still a big project but compared to a lot of wartime expenditures it is pretty modest, especially if the weapon could prove decisive or, as they feared at the time, would be necessary to counter a potential German weapon.

It turned out to be a very wrong estimate. The final cost was $2 billion, which is to say, some 5X what Bush quoted Roosevelt. It took 2.5 years to complete and almost wasn't ready in time for use in the war (if it had dragged on a few more months, the war probably would have ended without nuclear weapons being used). And as an aside: It actually took many more people than 130,000. 130,000 was the peak employment. Total employment was more like 500,000.

As for why a wrong estimate was approved, the decision was not aired widely. It was basically up to a few scientist-administrators (Bush, James Conant, E.O. Lawrence, and Arthur Compton). They were all pretty enthusiastic, having been convinced by the British MAUD Report that you wouldn't need that much U-235, and that they could probably cobble together a way to make U-235 pretty quickly. It wasn't vetted by any committees or anything. They got the President's ear, he said "OK," and that was that.

Which is to underline one point about the Manhattan Project that is often overlooked: its secrecy was core to how it got started in the first place. Had it gone through, say, a Congressional committee or something, the odds of it being approved would have been much lower. But it didn't. In fact, Bush deliberately avoided any outside scrutiny: as quoted in the page linked to above, when Bush went to Roosevelt, he explicitly noted that "It would be ruinous to the essential secrecy to have to defend before an appropriations committee any request for funds for this project".

As the project ended up dragging on, there were numerous attempts to audit it, by both Congressmen (including but not limited to Senator Harry Truman) and other federal agencies (e.g. the Office of War Mobilization) who wondered whether it was a good expenditure of money or not. Eventually the Manhattan Project leaders hooked a few Congressmen in on the subject so they would continue to approve their funding requests.

But that, and the vast increases in manpower and expense, were after the US had already committed a lot of resources to the project. And as with so many things, once committed to something big, it was easy to justify committing just a bit more if it looked like success was right around the corner. Which is just to say: once they were knee-deep in it, they felt the need to push it to completion, and one of the justifications for that (felt acutely by General Groves) was the fear that if they didn't, they would be called to court for their irresponsible spending. Groves once quipped to a subordinate that if they made an atomic bomb in time for the war, nobody would ever ask them about whether it was a good idea to spend that much money. If they didn't, nobody would ever ask them about anything else.