It's a question of whether the CHA modifier is a mechanic applied to the enchanter class, or to their charm spells specifically. The only way to be sure would be a lot of testing, though I'm sure all the shamans/druids/etc. with strings can provide some anecdotal reports.
The fact that those classes can use puppet strings effectively (presumably without carrying around a bunch of CHA gear) would suggest that the CHA modifier is something specific to the enchanter class itself.
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