Does the effectiveness of a harpy’s song fluctuate based on her charisma score?

The 5e harpy has a Luring Song ability:

Luring Song: The harpy sings a magical melody. Every Humanoid and giant within 300 ft. of the harpy that can hear the song must succeed on a DC 11 Wisdom saving throw or be Charmed until the song ends.

Would changing one (or more) specific abilities modify the harpy’s ‘Luring Song’ DC / saving throw? Are monster spell-like abilities (such as this one) linked &/or attached to specific ability scores in the first place?

Reason for asking: A harpy + bard would require changing basic harpy stats – does this even matter for her preexisting & innate song ability? Related questions obviously include examples like a Beholder (and their eye-rays) finding the right ability-modifying tome. And so on with creatures like Medusas (gaze), Dryads (charms) and Angels (spell-like abilities) with ability jumps and pumps.

Note: this must be in the monster manual somewhere – or even in StackExchange already. My apologies in advance for failing to find it.