According to this video, gassot have only been around for about a century. I know for a fact that when my father became bar mitzvah, his grandfather got him tefillin dakkot (his father got new tefillin for himself). When did it become common in the Orthodox world for people to purchase battim gassot for their tefillin?
Warning: This answer contains original research and anecdotal evidence.
I got curious about this because of how hard it is to open gassot and the risk of damaging the battim or parshiyos when you do. Gassot are currently the standard and some sofrim even refuse to sell dakkot, but the transition seems to have happened relatively quickly. A little over ten years ago, when I was getting my bar mitzvah tefillin the sofer showed my father and I both types of battim but we were told in school that "all tefillin these days are gassos". Today, I almost never see tefillin dakkot that were purchased less than 20 years ago; they are all either gassot or low-budget peshutim.
As an aside, I only watched the first minute or so of the linked video, but the "dakkos" that the narrator uses to demonstrate how flimsy they are are peshutim, not dakkos. As an aside to the aside, it is evident from the mishna that tefillin peshutim were common in the period of the tannaim.