What is the authenticity of attributing the aforementioned statement to our master Amr ibn al-'As, in which there is disparagement and portrayal of the two esteemed Companions as seekers of worldly gain, and if authentic, what does it mean?

The report mentioned about Ibn Asakir is weak in its chain of narration and cannot be relied upon, because its narrator, Ibn Ishaq, is a mudallis (one who practices tadlis in narration) and did not explicitly state tahdith (direct hearing), and the person who conveyed the report is unknown.

As for the reason for the disagreement between Mu'awiyah, Amr ibn al-'As, and Ali, may Allah be pleased with them, it was their demand for the blood of Uthman and the execution of qisas (retribution) upon his killers. This was an act of ijtihad (independent reasoning). Muslims agreed that Uthman was unjustly killed, and his killers were in Ali's army and possessed power. Ali was unable to defend Uthman, and for this reason, Mu'awiyah and Amr saw fighting as a means to repel their injustice.

As for Amr's pursuit of leadership, it was an act of ijtihad on his part, based on his perception of his own qualification for it, given the strength of opinion and resolve that Allah had bestowed upon him, and his experience in leadership. This does not contradict Islamic law, as a person may seek leadership if he knows himself to be capable of establishing truth and justice, as Yusuf (Joseph), peace be upon him, did.

What indicates Amr ibn al-'As's ijtihad and his not deliberately committing forbidden acts is what he said upon his death: "Then we became involved in matters regarding which I do not know what my state will be." This indicates his fear and piety.