I would not say that they have to precede a noun, since that implies that *"The should give you money", because "the" precedes a noun. Articles have to be followed by something in the NP, and what immediately follows could be all sorts of things ("the old cow", "the three dogs", "the over the hill gang"). Articles are not strictly only found before another word in the NP, though the contexts where they aren't is quite restricted. You can say "I said 'You can have these cookies, not the. Leave those ones alone". I'm sure some people will feel that these are unnatural, and they are clearly uncommon, but they are possible. In contrast, if you want to contrast bound-morpheme selection, you can't say "I didn't say [s]", or anything like it, meaning "I didn't say cats (I said cat)".
Under the assumption that a bound morpheme is definitionally "a morpheme that is always a subpart of a word", then setting aside isolation meta-linguistic uses of articles, they still are not part of the following word – unless you definitionally declare that they are. It is plausible to say that they are clitics, and there is a theory of clitics that they are syntactically separate words which are morphologically or phonologically part of some other word. See work by Phillip Miller from the 90s on the notion of "edge inflection", and his treatment of genitive 's in English.
So one theory would simply be that articles have a specific left edge location in the NP so that they can't be preceded by anything (at the NP level), and the rest of the NP has to contain something. Or, articles could be the realization of an abstract feature that is assigned to the left edge of the NP and is realized as a prefix morpheme on whatever is "really" the leftmost word in the NP. Under the latter construal, you could say that articles are bound morphemes. IMO it depends on your theory of clitics.
Also, one never really knows how many NPs an NP contains. There are quantifying expressions that come before articles, like "some of the people", "three of the cars", "all of the dog" and the latter can be reduced to "all the dogs". How you dispose of such cases is rather theory-dependent. They could be NP inside NP, or they could be special things that come at the left of the nounish-phrase.