I’d say that F. Jentzsch (1925, p. 863) gives just the concise summary you want (emphasis added):
Secondly, one had to understand and sufficiently clarify certain general properties of optical systems, such as the concepts of focal distance, focal points, principal points, magnification, etc. This was provided by Gauss. He was the first to bring his forerunners’ works to a certain closure, mainly by also treating systems with arbitrary thickness, and by showing that any (rotationally symmetric) system could be replaced in thought by a single refracting surface, or by a thin lens and a translation.
(“Forerunners” acknowledged in Gauss (1841, 1843) are Cotes, Smith, Euler, Lagrange, Piola, and Möbius.) A more detailed account, by C. Schaefer in Gauss’ Werke (Vol. XI.2, Abh. 2, §III, 1929), mainly agrees and adds (pp. 189, 201; paraphrased translation from Dunnington, 1955):
According to his own claim he had possessed the results for forty or forty-five years, but had always hesitated to publish such elementary meditations. A work of Bessel on the determination of the focal distance of the Königsberg heliometer gave him the impetus necessary for publication. Bessel’s method assumed mistakenly that the usual lens formula
is correct for lenses of finite thickness. As a consequence of this mistake, Bessel greatly underestimated the possible error of his measurement. Dioptrische Untersuchungen [gives] formulas for a simple lens of nonvanishing thickness [$\ldots$] While Bessel estimated the error of his result at 1/75,000, Gauss showed that it amounted to 1/1,300.