Some good points added; and I didn’t want to type much on my tablet keyboard yesterday.
The big shield that incorporation adds is protection from debt incurred by liability. Obviously, for a one-person corporation (well, one person plus whatever board nominally directs the corporation), you’ll need to come up with collateral for bank loans.
However, as corporations grow, the need for “bank” financing can also be outstripped, which is one of the reasons that corporations start to meet their financing needs through the bond and equity markets. That is, they sell bonds for some financing or “go public” and sell some ownership in the company to willing investors through shares. (They can also obtain private financing by doing the same thing more or less privately with venture capitalists and other investors, one-on-one or in small groups.)
However, another aspect to the growth of corporations is that the scope of their property, offerings and projects can also increase, to the point where it would be a huge personal risk to individual managers to take on some projects, unless they were somehow shielded from the risk of failure.
This is one reason why the details of the question were somewhat absurd. Of course you could pay off a $10,000 debt if you seriously expected to be in business in the USA. (You wouldn’t seriously be thinking of starting a real business – more than a sidewalk lemonade stand – with less capital than you describe in your example.)
However, it might be realistic for you to grow your corporation into the million-dollar range, for example, and then take on a project (which you may realistically and confidently believe you can pull off), and fail. Maybe the consequences of failure could include liquidated damages to be paid to the owner for every day that you fail to meet a completion deadline. (This is pretty common in some lines of contracting.) If there was no protection for the owner and other shareholders to be protected from the catastrophic failure of “we will never meet this completion objective; the corporation fails right now”, then everyone would lose everything that they owned. This would make capitalism pretty unpopular among capitalists, and it would make construction and other contracts so astronomically expensive that it would grind progress to a halt.
Other types of liability occur when legislation changes after you have been in business, such as happened with asbestos and tobacco. (This is not about whether some individuals and corporations have suppressed facts about their own knowledge of culpability or criminal activity.)
The largest protections offered by incorporation go to shareholders, who are the “other owners” in most corporations. I don’t have enough knowledge of all incorporated businesses in the USA (much less “in the world”) to say that “most corporations are owned by multiple shareholders”, but I think it’s fair to say that “the value of most corporate assets is owned by multiple shareholders”. If I had to be personally responsible for every action that General Electric performs, for example, then how likely would I ever be to purchase a share of GE stock – or any other stock – and how likely would corporations grow to the size that they can achieve great things?
So the shielding that incorporation affords owners is from “catastrophic” failures of the enterprise, not the garden-variety type of debt that you offer in the example.