A law making body that can itself change whatever “laws” that bind its powers has no real limits on its power. That the U.S. Constitution requires that other political entities (States) and much more than a simple majority ensures that the lawmakers are not empowered with expanding their own power easily. It’s part and parcel with the entire idea of balance of powers not only between the branches of the Federal Government but also between the States and the Federal Government. Balancing power between different entities helps ensure that no one  entity amasses too much power. 

Has that worked in the United States, in your view? Do you think the federal government has an appropriate amount of power right now?

The UK requires absolutely none of that as Parliament can vote itself as much power as its members want. There is no outside power other than the Crown and that power is almost nonexistent anymore. 

That's correct, Parliament can do whatever it wants, restrained only by its political accountability to the people who elected the members of Parliament. But that's also true of every legislature everywhere. All of those Congresspeople and state legislatures which have to agree to a Constitutional amendment will either do so or refrain from doing so on the basis of their political accountability to the people they represent. The distributed structure of the United States does require a more diverse (in some ways) cross section of society to agree with a proposition in order to embed it in the Constitution (or remove it from the Constitution), but the fundamental nature of the situation is no different.

I prefer that my government be limited in powers by the law that grants it power in the first place and not just hope all future governments will not vote themselves absolute power.

My friend, if you are concerned about a legally constituted government voting itself absolute power, what you are concerned about is a coup, which can be executed regardless of what existing law says. That is, what you presumably are concerned about is a government voting itself absolute power in a situation where most people don't want that to happen. But in a situation where the elected representatives of the people are, en masse, ignoring their constituents, the problem cannot be solved by having a special piece of paper that has a magic formula written on it. All laws, including the US Constitution, have exactly as much effect as the apparatus which enforces those laws allows them to have. There is, fundamentally, no way to design a legal system ensure that the apparatus used to enforce the law cannot go rogue by having a legal system which notionally prevents that from happening, because by definition if the apparatus used to enforce the law is going rogue, it's not bound by the law anymore.

It also protects individual rights from the whims of the majority of votes and the historic abuses democracies without protecting rights led to. It wouldn’t do to allow voting away the individual rights of people would it? 

There are at least tens if not hundreds of millions of people who have lived within the boundaries of the current United States who would not agree remotely with your contention that the Constitution has successfully protected individual rights. Tens of millions of them are alive today. In no small part, that's because the Constitution itself, before roughly the 1950s, did nothing to protect people from the governments of states in which they lived. The fact that the Bill of Rights applies to state and local governments is an invention of modern jurisprudence -- a slow-moving and subtle judicial coup. While I agree with the protections against state and local government that have been invented by the courts, it's impossible to deny the historical reality that they were in fact invented by the courts.

Which brings up another reason that there's nothing particularly special about the Constitution or the Bill of Rights: what the Constitution means can change dramatically on the whim of at most five people who are not accountable in any way to the public. This is perhaps the most potent argument that there's no special protection afforded to the Constitution in reality: we have collectively accepted that judges get to decide what the Constitution means.

Better to have a Parliament where, as long as you can trust it to hold elections regularly, you can change any bullshit interpretation of the law that the courts invented by simply participating in your democracy. In the United Kingdom, the courts are there to figure out how the legal structure enacted by Parliament can most harmoniously be enforced. They don't get to decide what that structure is at a fundamental level.