Welcome, J Li
It isn't strictly true that Kant focuses on a relatively small set of principles when discussing morality.
It is true that in the Groundwork of the Metaphysic of Morals (1785) he discusses only four examples to illustrate and tentatively vindicate his ethical theory. But Kant's discussion of morality is by no means confined to this brief text. In the Metaphysics of Morals (1797) he ranges over a broad variety of ethical topics - moral examples - from commerce and property, marriage, friendship, humility and hypocrisy, lust, money, murder, prostitution, gambling, punishment and public executions, revenge, and the distribution of wealth - among many other topics.
So why only four examples in the Groundwork- the ban on (1) suicide and (2) false promises, plus the requirement (3) to develop our talents and (4) at least on occasion to help others?
Kant is not a revisionary ethical theorist, unlike (say) the early utilitarians such as Bentham. He takes morality as he finds it, or as he believes it is, and undertakes to show that his categorical imperative test as applied to maxims produces results that agree with ordinary moral thinking. He takes such agreement as criterial for the correctness of his ethical theory.
The criticism has often been made, and rightly, that ordinary moral thinking, even in the form in which Kant conceived it (heavily influenced by German Protestantism), was more flexible about making a false promise than Kant recognises in the Groundwork.
The major point is, however, that Kant needed to test his ethical theory against ordinary moral thinking, since he was trying to theorise morality as it was or as he believed it to be. He took it to involve at least four prohibitions as basic - to oppose suicide, making false promises, failing to develop one's talents, and never helping others. Applied to the maxims of one's actions, the categorical test precisely (as Kant thought) ruled out all four prohibitions of ordinary moral thinking.
He regarded this as strong prima facie evidence that he had in his ethical theory, with its doctrine of the categorical imperative, captured cornerstone requirements of ordinary moral thinking. Remember, the Groundwork of the Metaphysic of Morals was just that - only a Groundwork. The heavy duty work of a comprehensive ethical theory was reserved mainly for the much later Metaphysics of Morals.
I. Kant, Groundwork of the Metaphysic of Morals, tr. M. Gregor & J. Timmerman, rev. ed., Cambridge: CUP, 2018.
I. Kant, The Metaphysics of Morals, tr. M. Gregor, rev. ed., Cambridge: CUP, 2017.
O. O'Neill, Constructions of Reason, Cambridge: CUP, 1989.