Have you ever wondered how the master of the house who is leading the Pesach Seder can provide Matza for all participants at the table from his 3 Matos? It is clear from all the Halachic literature that no one at the table but the master of the house had Matzos. The Pesach Seder was no different to any other Chag or Shabbos, where the master of the home makes HaMotzi and hands out the bread.
Furthermore, Halacha instructs us to make the middle Matza, which is to be torn apart [broken if using hard Matza] larger than the others Matzos, since all the participants at the Seder will require a Kezayis from that smaller part.
Today's matzos barely have 3 Kezeysim total, the torn middle Matza will not yield 2 Kezeysim.
Our traditions also describe the three Matzos as being made from one Isaron of flour (about 2Kg, which is the Shiur that requires that Challah be separated); each Matza would weigh about 750 gm. Such Matzos could only be made as thick Matzos. Skinny Matza would be too large to place on the table and would certainly break under their own weight.
RMF writes that the recent practice of each male participant at the Seder having their own KeAra is due to our tiny Matzos.