Here are six answers to the question in my heading. Arguments can be made in favor of each.

1. The founding took place from 1776-1789, from the Declaration to the Constitution. Although the architects of the new republic sought to recycle some existing materials, they drew a new blueprint for the base on which our system stands. It was designed to be alterable–and it has been altered–but the foundation is still recognizable.
2. The foundation was poured in 1492 and 1619. Once Europeans began seizing land from indigenous people and importing enslaved Africans to work that land for them, the basic arrangement was set. The 1788 Constitution essentially preserved that structure. Some better things have been built on top of it, but the original floor is still down there, not far below the surface and determining what can be constructed above.
3. The social contract has been renegotiated at several key points: 1776 (Declaration), 1781 (Articles of Confederation), 1788-9 (Constitution), 1864-5 (post-Civil War amendments), 1938-45 (the Supreme Court reverses itself and allows the welfare state), 1954 (Brown v Board), and arguably again since the 1980s. These shifts are fairly fundamental and not well described by treating the 1788-9 contract as still foundational. The 21st-century political system is incompatible with the Framers’ plan, but that is because we have chosen to lay new foundations.
4. The United States was founded in 1788-9. In the 1900s, we ignored some of its basic principles, such as the list of enumerated powers, without explicitly and legitimately renegotiating them. The foundation is still in place, but we have built unsound structures on top of or beyond it. The Constitution is “in exile” (a phrase apparently more used by critics of this view than proponents of it).
5. The actual political-economic system in which we live is fundamentally based on publicly traded corporations, industrial production, organized labor, regulatory agencies, credentialed professions, public and private bureaucracies, mass media, mass schooling, securities markets, electronic networks, science (as a set of powerful institutions), databases of people and objects, and a permanent war machine. These elements are not envisioned in the US Constitution, which influences them somewhat but hardly determines them. The same basic structure is evident, for example, in Canada. The modern foundation has been poured one layer at a time, but if you had to pick a symbolic date for this option, it might be 1908, when the first Model T rolled off the assembly line.
6. There is no foundation. A society is not well understood as a base and superstructure or as a single game with basic rules. It’s a complex, emergent system best understood as whole series of overlapping and interacting institutions, each with rules of its own that affect the other institutions’ rules. All is flux.

I realize that most people don’t explicitly discuss this question, yet I think that today’s opposing ideological camps would each answer it differently. It could even serve as an ideological Rorschach Test.

See also: constitutional piety; the Citizens United decision and the inadequate sociology of the US Constitution; the role of political science in civic education; polycentricity: the case for a (very) mixed economy.