This move has always confused me due to how many Pokemon can learn it that aren't Grass-types, as it involves controlling plant life

For the longest time, that confused me too. As all the flavor messages seemed to point that a knot would be made controlling grass onto which the foe would trip over.

But then I noticed that it makes contact. So it isn't remotely manipulated, summoned grass; contact means the user will be actually close and maybe even touching the opponent.

Yet that manages to only make things more complicated: More than stealthily setting it up for the opponent to trip it could only work as a grappling throw (specifically, a leg hook). Which kind of makes sense despite being close to Fighting as other general and specific fighting techniques are branded with other elemental powers from its origins...

However in this case it comes from... employing a grass rope. With actual, physical grass. Other moves, such as Aerial Ace or Bullet Punch, take their typing more from theming, and even then are less widespread outside their typings...
But here, the pokémon, regardless of their type, can use specifically a grass rope and that alone makes this move grass typed.

But not only that, but it is Special. Now, Special moves that make contact make perfect sense, since they might require proximity/touch yet still attack with mystical propertypes, which is probably how Petal Dance works. Yet I'd think something as... physical as a leg hook or even actually remote tripping would be, well, Physical. (I even thought for a long time the move was Physical, until I actually saw its status screen)
The best explanation I can think for this is relating it to the Aikido martial art's philosophy that blends the spiritual and physical... but this is way closer to what the Fighting type represents. And while it could easily have flavoring relating Ki to plant life (heck, some real techniques do relate specific ways to manipulate Ki to plants) which would explain the scenario of a "grappling contact-special-grass move"... in this case there's no such flavor.
It is merely tying the opponent with grass which a really wide variety of pokémon have access to for some reason.

So yeah, no matter how much I overthink it, I still don't get this move.

-- Dragon Dance

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