I know that the halakha requires n'ttilath yadhayim before the recitation of sh'ma, before t'phila, and before and after eating bread (cf. Hilkhoth B'rakhoth 6). And I also understand that n'ttilath yadhayim requires a k'li and koah nothen (and clean, unused water etc., ibid.), but my question is if clean hands is the goal, then why wouldn't washing one's hands under a faucet work? What is the point of all of the particulars listed in the halakha if not to ensure a useful amount of clean, flowing water that then ensures clean hands? Doesn't a modern faucet effect that in the same way as - and in many respects better than - n'ttilath yadhayim?
Rabbi J.B. Soloveitchik (Reshimot Shiurim, Berakhot p. 202) suggests that whether one requires a vessel for hand washing prior to bread depends on a dispute between Rambam and Ra'avad in Hilchot She'ar Avot Ha-Tum'ot 8:8. Ra'avad there extends the law of washing hands for ritual purity to all hand washing, but Rambam implies that this reason extends only to terumah. Therefore, according to Ra'avad, one must use a vessel when washing for bread, because the reason is not just cleanliness, but removing ritual impurity of the hands. But according to Rambam, it seems that a vessel may not be necessary. Furthermore, Rambam in Hilchot Berachot 6:2 seems to equate hand washing prior to bread with hand washing for prayer; in this case, when one has no vessel one may use any substance that cleans (Berachot 15a).
The Arukh Hashulhan discusses this issue in detail in Orah Haim 4. Your assumption is indeed correct; when clean hands are the only goal, such as when you scratch your head or the like, using a faucet or other flowing water would be fine, if not preferable to using a vessel. However, the netilah prior to eating bread and before shacharith involve other considerations. Washing before bread was originally instituted for kohanim only and was simply extended to others so as not to differentiate between people. Washing in the morning is due to cleanliness as well as to get rid of ruah ra and because we are considered new creations upon waking after a night's sleep. It is disputed amongst the Rishonim what exactly is required in these cases, and what suffices when one does not have access to the ideal resources.