It's an interesting question, but I'll have to start from the theory before I get into the practical realities of the U.S. system of Common Law.

There's a variety of philosophies regarding what the 'law' ought to be. I'll be brief and say that there are two that are taken the most seriously: Positivism & 'Natural Law' -- both have a variety of subgenres, but they both provide two of the instincts as to what the law ought to be. 'Positivism' follows the idea that the law needs to be -- first and foremost -- entirely predictable & clear to the people who are in it's jurisdiction. To this end, you want to either have extensive, but plainly written legal codes which account for a variety of situations and clearly establish liability in various cases. It would also, necessarily, strongly prefer rules established by 'precedent' -- so that when new legal area is charted because the current rules weren't clear enough, everyone can cite this new ruling as a reliable source of information about how the law will be going forward. Positivism makes few claims to ethical or moral superiority -- laws are explicitly not moral guidelines, ever -- under Positivism, and ethically there's usally just the overarching impression that it's important for any society to have maximally clear & reliable laws so that people can know what obligations & duties they have to each other. If that clarity reveals facts that the citizens dislike, Positivism's parting gift is that the rules for changing the law ought to be clear as well.

'Natural Law' on the other hand, is a much more explicitly moral system, and it's certainly reserved for moral realists (as opposed to moral anti-realists, people who think that 'moral facts' don't exist). It's a whole big thing in ethical philosophy, but I'll confined my comments to philosophy of Law. Natural Law theorists believe that the point of the Law is to conform as closely to rules of morality as possible, whether those rules were laid down by God, Reason, or whatever their moral philosophy says concretizes abstract moral facts into the material fiber of our reality. To that end, if a Law is demonstrably unjust, then it has no place in the system and ought to be discarded as rapidly as possible. When laws are drafted and put into practice, it's always with the goal of shaping society/the Law into a form that conforms to morality. One idea that gets thrown around a lot is that "Law is the Perfection of Morality" -- with that lovely 17th century capitalization -- and that as our understanding of moral reality grows, so to does our approach to the law have to grow and change. This makes notions of 'precedent' a bit hazier, because relying on the law at a time when it was morally wrong would obviously not be valid. there are plenty of in-built protections for, say, not holding people liable for doigns things that they didn't know were crimes when the law changed, but the thrust of this theory is that the Law should never tolerate immorality & grows with our understanding of what ought to be. So, Judges, or whoever is responsible for clearing legal matters, would have an obligation to the 'the right thing' rather than be bound by precedent or procedure. Of course, as always, it's all a balancing test , because to some extent people do like clarity and procedure. It's not like this theory imagines we'll be having a bunch of Judge Dredds running around and forcing ships to adopt radio technology in the thirties.

But now we can wrap around to your original point -- why Common Law? The answer is because it lends itself to predictability and clarity. Common Law, broadly, refers to codifying legal matters as they were practiced by the people, as well as respecting legal precedent. So when a bunch of English peasants thought that both sides needed to trade valuables for a contract to be valid, the idea of 'consideration' was born and blah blah 700 years later we now have the Uniform Business Code that very clearly lays out exactly what you need to do a bunch of business stuff -- some of which derives from these old legal concepts. For all that it sucks to be beholden to old ideas -- the advantage is that you know exactly what magic words you need to utter or write down to make the courts do what you want them to do. You can look up what prior courts have done and force courts to agree with you based on precedent! So you can think of the Common Law system of favoring positivism over natural law -- although, obviously, a system this large is going to have hybrid urges. Furthermore, at least in the U.S., there's plenty of legal codes and legislation that are wholly new and totally override 'old' legal concepts. It also depends a great deal on the field you're in. Laws surrounding real property are by far the most 'feudal' and cling to a lot of old concepts -- but something like insurance law or even the Criminal Code? That's all a creature of modern legislation, although 'modern' stretches back to the 1940s sometimes.

I'm not sure if your problem is primarily with relying on 'precedent' generally, or if you think that over-reliance has allowed concepts that are unfit for a modern society to creep in from older times. I think that's an important distinction to make, because precedent on a whole is profoundly important to a functioning legal system. The king changing his mind day by day and giving different rulings to the same problems? Not that's some feudal stuff. Having a Worker's Council that will tackle problems differently depending on the mood of the participants that day might be objectionable to the people who are bound in their jurisdiction.

That's really all Common Law is -- precedent. While some concepts from Ye Olde Englishe Lawe have been retained, most of the particularly objectionable ones have been filtered out by almost a thousand years of people arguing with each other. If you have some particular disagreement with a stupid remnant of tradition that has lodged itself in the legal system, then I might be able to explain it in a way that makes it seem less stupid.

Source: am a lawyer, and I want to kill myself every day (but that's because of the shitty people I work with, and not because the law is fundamentally flawed)

edit: Check mrhellblazer's post about the good that can be done with common law and precedent for a more detailed defense of the concept