It was found in Peirce's unpublished papers that he defined the NAND operator and discovered that all classical logical operators could be replaced by combinations of NAND. What are some possible motivations for Peirce's use of only one operator(his NAND) to recreate the "and", "or" and "not" operators?
See Charles Sanders Peirce, Collected Papers : Volume 4. The Simplest Mathematics (1933), page 13:
[4.12] A Boolian Algebra with One Constant [untitled paper c.1880]
Every logical notation hitherto proposed has un unnecessary number of signs. It is by means of this excess that the calculus is rendered easy to use [...]; at the same time, the number of primary formulae is thus greatly multiplied, those signifying facts of logic being very few in comparison with those which merely define the notation. [...] The apparatus of the Boolian calculus consists of the signs, [...]. In palce of these seven signs, I propose to use a single one.
Two propositions written in a pair are considered to be both denied.
And see [4.264] [from The Simplest Mathematics (1902)], page 215:
for example x⋏y [a sort of "harpoon" symbol] signifies that x is f and y is f. [...] With these two signs, the vinculum (parentheses, braces) and the sign ⋏ , which I will call ampheck (from Ancient Greek ἀμφήκης, "cutting both ways") all assertions as to the values of quantities can be expressed.
Actually, my slightly snide comment above may be close to an answer. All possible logical operations can be produced with some combinations of NAND operations, meaning that only that one operator needs to be studied. While I have no idea how this would "simplify" philosophical problems, it would at least regularize the notation and perhaps make contemplation easier for folks with a "mechanical" thought process.
This certainly made early digital electronics simpler to deal with.