If you recall Tswett's original discovery of chromatography, he used a calcium carbonate column to separate the chlorophyll pigments (or different types of chlorophylls). Now calcium carbonate is a very fine powder and the permeation of liquid through a powder is very slow, especially if the column is long. Pressure drop across a column scales as 1/($d_p^2$), where $d_p^2$ is the particle diameter.
Paper chromatography was well known in the 1930s, it was relatively faster but you are limited by choice of stationary phases, just cellulose. In order to have "paper chromatography" with other stationary phases such as silica, alumina etc., a thin layer chromatography was invented. The main purpose was speed of separation for quality control of pharmaceutical products, as it is today!
So this "thin" in TLC is akin to a thin paper sheet, but made of different stationary phases. Note that if you use a thick layer, the capillary action will not be uniform, and the bands of analytes will be distorted badly.