Article One of the Constitution assigns to authority to "borrow Money on the credit of the United States" to Congress. Modern constitutional interpreters, especially those who sit on federal courts, tend to give Congress a lot of leeway when it comes to delegating its assigned duties. But even this leeway has its limits.
The current arrangement for issuing debt, which originated during the Great Depression, is probably at that limit. The issuance of debt is largely delegated to the Treasury, with Congress just setting the upper limit of the debt. Further delegation of this legislative power to the executive is probably unconstitutional. (For more on the history of debt issuance, see here.)
This is not just a fetishization of the Constitution. Even if we were starting from scratch, we'd want to incorporate a requirement for Congressional approval of debt issuance. Congressional approval makes defaulting on debt far less likely than it would be if the executive branch could issue debt on its own. Debt holders can rely on the fact that the debt really is backed by the full faith and credit of the United States, something that would be in doubt if Congress did not have to approve the debt level. Because of the debt ceiling vote, the debt is never just the "Obama's debt"—it's debt that the people's house approved.
The debt ceiling also holds Congress accountable. If raising the ceiling were left to the President, more and more Congressmen would vote against raising it because a vote against would be cost free. What lawmaker wouldn't want to cast a purely symbolic vote for "fiscal responsibility?" The current arrangement makes this vote count for something, which means that lawmakers must seriously consider the costs and benefits of their position on the ceiling.
Finally, it is worth remembering that payments on the debt must be authorized by Congressional acts. Absent a debt ceiling, Congress would find itself asked to authorize payment on debt it had not approved. Under Geithner's proposal, the debt ceiling could be raised against the wishes of a majority of lawmakers. This majority, however, could still vote against the authorization of payment on debt. The very threat of a payment default vote would be far more unstabilizing than the debt ceiling debate has ever been.
I get the feeling that a lot of critics of the debt ceiling just don't understand it. They're like the walker in "Mending Wall," Robert Frost's poem who comes across a wall between farms but can't understand the reason it is there. Before kicking over the wall, it's at least worth investigating the reason someone thought it was worth putting up in the first place.