Something that may be also helpful in thinking about this issue is to consider to what degree accent is contrastive in classical Greek and to what degree it is simply a feature of the prosody.
So regarding the point about "a single word pronounced two different ways" -- it seems to me this is only weird if a change in pitch accent would automatically also affect meaning (In which case you're maybe not talking about a pitch accent anymore, but about lexical tone). I'm not sure that's the case with Greek. Obviously classical Greek does have places where a difference in accent is connected with a difference in meaning, but this is more the exception than the rule, mostly it seems to be pretty mechanical (recessive accents in verbs etc) & the accent changes with two-syllable enclitics is also purely mechanical. So I feel like functionally accent in classical Greek has more to do with word and phrase boundaries than it does with semantics.
In the case of enclitics, there's also the fact that words are being "run together", similar to what happens with elision or contractions, so I don't know if you can really validly argue that it's just "one word being pronounced with two different accent patterns" -- rather, it's one word being pronounced differently than "usual" because it's part of a phrase. In English we can vary where the stress falls in a phrase such as "it is not" ("it's NOT" vs. "it ISn't"). Is the Greek phenomenon any stranger?
Building on this, especially if you consider the alternation of acute/grave in the ultima depending on whether the word is at the end of a clause or not, I think it's maybe also important not to ignore the fact that intonation patterns in a lot of languages are dependent on the structure of phrases, i.e., where the important words are in the sentence, whether it's a question or statement, etc. I realize that many of these are going to be languages with a stress accent, not a pitch accent, but I'm not sure it's possible to draw a rigid distinction between intonation and accent (regardless of what type) because they always work together.
IPHIGENIE: Kann uns zum Vaterland die Fremde werden?
ARKAS: Und dir ist fremd das Vaterland geworden.
IPHIGENIE: Das ist's, warum mein blutend Herz nicht heilt.
(Goethe, Iphigenie auf Tauris)