First of all, it's worth remembering that spoken languages aren't designed by committee - they change over time, and not always following a strict logic.
But the reason for the change may have been as follows:
Start with the fact that in lashon Chazal (which, again, represents the spoken and written Hebrew of their time) there's a tendency to leave off the ית- at the end of a fraction. Biblical שלישית = Chazalic שליש; Biblical רביעית = Chazalic רביע. Could be that this is because the forms with ית- are ambiguous: שלישית can mean "(fem.) third (in a series)" or "one-third."
With ששית you can't do that without introducing another ambiguity, because then you're left with שש, which is the cardinal number "six." (The difference in nikkud probably wouldn't have been enough to resolve it.) So the logical thing to do would be to borrow an Aramaic-derived form instead.
There's also the complication that we do find Chazal often using the term רביעית, and sometimes also (as mentioned in comments) שמינית and עשירית. It might be that רביעית had already become established as the name of a unit of measurement, and it kept that name even when the term for the fraction was changed. In fact, רביעית generally is used just for that specific measurement, a quarter of a log; as a fraction (or a different unit of measurement, such as a quarter of a kav), the term רובע is usually used instead. (In that sense it might be a little like the way in English we have "quart" as a measurement vs. "quarter" as a fraction.) And so the related שמינית - which is half of a רביעית and was probably used fairly commonly too - also kept its original form when referring to an eighth of a log (or its metaphorical equivalent, like אחד משמונה בשמינית).
(As for עשירית - that seems to be used only when talking about עשירית האיפה, the measurement for a mincha, where that's the expression used in Chumash for it. Keep in mind that the eifah itself was no longer in use as a day-to-day unit of measurement in the times of Chazal. In other contexts the word עישור tends to be used.)