The original was probably a “th”, and the evolution into “T” (among Sefardi) and “S” (among Ashkenazim) occured well before modern times. The dialect spoken in Eretz Yisrael evolved naturally and combines all the mistakes of the leading dialects, which is logical since most languages evolve that way (that Israel Hebrew fails to distinguish between Aleph/Ayin, Saf/Taf, Daled/Thaled, Gimmel/Jimmel, kawmatz/patach, non-use of the thrid person future feminine, use of direct word order, etc.). Spoken languages tend to simplify over time (e.g. English having lost almost all use of gender, subjunctive, and the loss of its gutterals).
This is a fascinating subject for linguists, but of no theological importance.