The crux of my question is, why does there need to be a fancy handling of this particular sort of statement?
Because English lost a whole lot of its verb inflection, so certain shades of meaning that could once have been expressed through morphology are now expressed through syntax instead.
Why is the list of modal verbs so limited?
Because they're function words rather than content words, and function words tend to be closed classes (you can't easily make more of them) and very limited in number.
And why do they, in English at least, reuse an existing verb, giving it a totally different meaning?
In English, how do you go from "to have" (possession), to "I had climbed the tree".
The "have"-perfect is a feature of the Standard Average European sprachbund, which means a whole lot of languages in the same area have this feature. It seems to have originated in Latin, persisted into the Romance languages, and spread from there into non-Romance languages like English and German.
Back in Latin, there was a single verb form (scripsī) used for two different tense-aspect combinations: past aoristic ("I wrote") and present perfective ("I have written"). This was somewhat ambiguous.
So over time, a new construction arose for the present perfective, using the verb habēre "have" with a participle. This probably arose through an evolution like:
- (Litterās scriptās) habeō "I have (written letters)" (as in, I am holding letters, that have been written)
- Litterās (scriptās habeō) "I (have written) letters" (as in, the letters have been written, whether or not I'm holding them right now)
This construction then became very prominent in Romance in order to distinguish the present perfective from the past aoristic, and spread from there to other languages in the area. English "have" is etymologically unrelated to habeō, but it looks very similar and has a similar meaning, so it was the obvious choice when borrowing the construction.
If I were to try and break this perfect tense down a little bit, I would do something like…
Look into "aspect" as a distinct thing from "tense". Perfective aspect ("have" in English) indicates that something was completed before the time you're talking about. If I say "I have eaten", I'm talking about the present state resulting from a past action of eating.