Actually, the corporate veil existed long before the LLC (which is a specific kind of corporate entity). I believe the first law that established the "limited company" was made in England in 1855.
The LLC is a structure that allows people to pay taxes on their owner's draw at a personal rate rather than pay corporate income tax and then be taxed again on dividends. The LLC is limited in the number of shareholders it can have. The LLC was designed to make partnerships more practical as partnerships typically had no limited liability (except those new-fangled LLPs that are basically an LLC anyway) The basic idea was this:
Partnership structure + Corporation's liability protection = LLC
LLCs make it easier for people to start businesses, simplify taxes and reporting requirements. They are a very good idea, and the limited liability part simply protects owners and employees from personal liability for things like leftover debt when the company goes out of business. The "corporate veil" can be pierced if members (shareholders) intentionally do illegal things or do unethical things like co-mingle personal and company money.
This whole "corporations are bad because they have limited liability" meme is not interesting. The limited company was invented to allow businesses to fail without taking the owners down with the ship and also to prevent companies from using their owners as an unlimited checking account. Imagine owning 1000 shares of, say Circuit City and having to pay $10,000 to pay your share of Circuit City's liquidation debt. Investors would be limited to the ultra rich, and middle and lower class people would be unwise to invest in stock... because company managers would know that ultimately, any check they wrote would be made good on the backs of the shareholders. I suppose we should go back to the pre-1860s economy where we had debtors prisons (yes, US States had them up until the 1850s) and where a business mistake or a recession could literally result in families being totally destroyed.
Talking about how having owners not engaged in managing the company (Adam Smith's big concern) is dangerous is much more interesting.