The central teaching of חכמה ומוסר ח״א׳ — מאמר ב׳: נושא בעול עם חברו is that nosei b'ol im chavero is one of the very foundations upon which Torah acquisition rests — not a natural emotional sympathy, but a deliberate intellectual act of contemplating one's own sufferings and transferring that understanding fully to others — and that it stands as the comprehensive expression of all virtues, which is why it must precede Torah study itself.
חכמה ומוסר, חלק א ב makes the connection to kingship explicit: whoever bears his fellow's burden more completely is more fitted to be king, for the essence of kingship is tending to the needs of the entire community, watching over every injustice, and rescuing the oppressed from the hand of the oppressor — so that kingship is, at its root, built upon nosei b'ol im chavero in a wondrous degree.
This is why, Chochma uMusar 1 p. 8 explains, David and Solomon desired kingship — not out of pursuit of honor, but because kingship is the supreme vehicle for carrying the burden of the entire world together with all people, making it the highest expression of this quality and the root of all worldly rectification.
חכמה ומוסר ח״א׳ — מאמר ב׳: נושא בעול עם חברו illustrates through Moshe Rabbeinu — who went out and saw his brothers' suffering through deliberate contemplation, not mere instinct — and through Yosef, whose practice of this trait led directly to his elevation to kingship over Egypt, that it is precisely this quality, pursued through the intellect, that qualifies a person for leadership over Israel.
חכמה ומוסר ח״א׳ — מאמר ד׳: איכות נושא בעול עם חברו extends the principle to show that nosei b'ol im chavero is the very foundation of redemption: Moshe and Aharon first had to earn the people's trust by genuinely sharing in their suffering before faith and liberation became possible, and even the smallest act of burden-bearing — as with Rivkah drawing water for Eliezer's camels — carries immense reward, while any failure to practice it entails catastrophic loss.
Chochma uMusar 1 p. 423 deepens the teaching further: one who carries his fellow's burden bears the heaviest possible load, and this obligates him continually to add more, as illustrated by Elazar the chief of the chieftains, whose very greatness meant his burden and yoke were greater than all others — the higher one's position, the heavier the obligation to carry.
The Alter's reading of the Yosef narrative in חכמה ומוסר, חלק א ב draws on the detail that Yaakov 'kept the matter in mind' upon seeing that Yosef was deeply concerned for the welfare of others beyond measure — this excess of concern for others being precisely the mark of one suited to kingship, which the Tosefta corroborates by describing the king as one who is occupied all his days with the needs of the community.
The Gemara in Sotah 11a — which describes how the Egyptians hung a brick mold around Pharaoh's own neck so that no Israelite could claim to be too delicate for the labor — provides the Alter's backdrop for understanding how even a corrupt king exploited the principle of shared burden as a tool of oppression, throwing into relief how genuine nosei b'ol im chavero, exemplified by Moshe's going out to see his brothers' labors (Shemot 2:11-12), is the redemptive inversion of that counterfeit.
רבינו ירוחם, דברים ב עמוד רלט reinforces that nosei b'ol im chavero is not external participation or mere sympathy but means being literally within the other's burden and pain — drawing on the verse 'in all their affliction He was afflicted' (Yeshayahu 63:9) — so that one actually suffers the same distress and carries the same weight as one's fellow, which is also the secret of 'taking one sixtieth of another's pain.'