Santoku means three (3) virtues, the santoku is utilized in aisan cooking, primarily for fish, meat and vegetables. The price is dependant on the range/quality of the knife. Upmarket santoku cuts through hard vegetables like carrots as if you are slicing through butter.
Much of what you say here is true in a rather limited sense. Let's be precise:
1) Santoku certainly means "3 virtues," but it is not clear which set of three virtues is intended; you will find at least four lists presented as gospel truth if you search hard enough. One set is a classical formulation of three Buddhist virtues. One set is a kind of modernist re-statement of those same virtues, such that thrift enters into it. One set is a list of foods (usually, but not always, fish, meat, and vegetables). One set is a list of things a knife does (chop, slice, mince is, I think, the usual set, but that's from memory). Since nobody that I am aware of has ever successfully tracked down the original advertisements that named this knife "santoku," it's all pretty much a guess at this point. Bear in mind that the same knife is equally commonly called bunka-bochou, which appears to be at least as old a name and has nothing whatever to do with three virtues.
2) By "Asian cooking" one presumably means Japanese, since it's not extensively present outside Japan.
3) Price is partly dependent on quality. With this knife more than most, outside Japan at least, price is also very largely dependent on advertising. A lot of folks have this general impression that Asian or Japanese knives are supposed to be better, and then the santoku gets pushed at them as the best thing ever. Since the market will in this case bear a lot of high prices, the prices are often grossly inflated.
4) A really sharp knife of almost any kind should go through a carrot easily. A santoku has no special virtues here over other knives appropriate for cutting carrots.