The closest we have to re-writing it is adding amendments

That's only a piece of the process. The United States is somewhat unique: we have a written constitution, but we have also assigned the judicial branch the authority to "say what the law is." This language comes from Marbury v. Madison. Although people might like to describe the courts as only doing interpretation of the law, it goes much deeper than that, to allow the constitution to adapt itself to changing times.

The best example of this "informal amendment" procedure happened during the New Deal. The constitution gives Congress the power to regulate interstate commerce and international commerce. When the first set of New Deal legislation was passed, the Supreme Court struck a lot of it down. It interpreted the power narrowly, only allowing congress to regulate interstate traffic of goods and services, the shipping industry, and things of that nature. This may or may not have been the interpretation the authors of the constitution foresaw, based on their expectations for the role of a national government in an economy where broad segments of production were intrinsically local and there was a clear distinction between the production and movement of items in commerce. But as the industrial economy began to produce goods for an increasingly delocalized market that disregarded political boundaries - as we began to enter the world where commodities are fungible - the Supreme Court effectively dismantled the distinctive between intrastate and interstate commerce. Later, it allowed broad delegation of rule-making discretion to executive agencies. This allowed, for example, the EPA to create rules to regulate anything meeting a broad definition of "pollutant" without a congressional vote on each specific rule. There was a time when this would have been questionably constitutional under the "non-delegation principle" that prohibited congress from passing its legislative responsibility onto other branches.

Together, these changes enabled the modern administrative government, where agencies in the executive branch can make environmental, safety, financial, and occupational regulations. This system had need been contemplated by the politicians who wrote, voted on, and ratified the constitution. For all intents and purposes, the constitution had been amended.

Another feature of the constitution is its use of intentionally ambiguous language. The fourth amendment prohibits "unreasonable search and seizure." The first amendment protects "freedom of speech." What does this language mean in practice? The courts decide. Effectively, what the authors did was to authorize the Supreme Court to set rules, about the police function, about which forms of speech society deems valuable, about what it means to be equal before the law... This strategic use of ambiguity can be found all throughout the constitution. As a law student, I spent basically an entire semester learning about free speech doctrine that came out of the court's having to dynamically make something out of the simple phrase "freedom of speech" as social values, technology, and norms have changed over history.

Another point about the judicial system: We are a "common law" country, in the English tradition. It is politically fashionable to talk about preserving an original intent of the founding fathers. However, when these people (many professional attorneys) were writing the document, this is not how courts worked - not in the colonies, and not in Britain. They saw courts as articulating a certain natural logic of society. The courts made explicit what was implicit in the given conditions of the nation's life. They believed that there were certain logical limitations on the power of government over citizen in a republic, and it was the role of the courts to find these limitations. This "due process" power has allowed a lot of ad hoc legal reasoning that conformed the idea of constitutional rights to the issues of each generation. This is how we have arrived at, for example, a right against certain prohibitions on abortion, or a fundamental freedom of parents to control how their children are raised (this was a controversial issue in the 20th century, when, for example, states passed laws preventing German immigrants from teaching their children in German), or the freedom to marry (including, but not limited to, same-sex marriage). This due process power has also been used to limit police interrogation techniques and other harsh applications of force against a person's body.

All of these combine to create a document that is much more, to use a trite expression, alive than people might think just by reading it.

My source for most of this is Chemerinsky's text on constitutional law, which obviously talks about the legal doctrines but also the social and political history of the Supreme Court's decisions in depth.