Yes, lots of age-old sleights will have French names in French texts. So assuming the sleight was first described or popularized in French texts, then it would be plausible that early English-speaking magicians and magic writers would have learned it directly or indirectly from these texts and adopted the French term.
And assuming they adopted the French term (in this case "le tourniquet") it would make sense to simplify it to "the French Drop".
Of course this is just a guess. If the sleight was described in English beforehand, then this theory becomes less plausible.