Not only did Moses tell God to relent/repent/turn from his anger, but God actually did so, according to the text. Verse 14 says:
KJV And the Lord repented of the evil which he thought to do unto his people.
NASB So the Lord relented of the harm which He said He would do to His people.
NABRE So the Lord changed his mind about the punishment he had threatened to inflict on his people.
I would not say that Moses thought his own leniency was a better approach than God's threatened punishment. Jewish tradition holds that Moses knew God's heart better than anyone in history. Christians might add "except Jesus." Like Abraham in Genesis 18:25, Moses' relationship with God was so intimate that he dared to contradict "the judge of all the earth," knowing that God's chesed - his unchanging love and mercy - underpins any temporary feelings of anger toward his beloved people.
Readers may interpret the story in various ways: a test from God to see if Moses truly understood God's will; a dialog in which Moses reminded God of God's own true feeling; or a story to remind the Israelites how blessed they were to receive God's unmerited grace through Moses' mediation. In any case, the lesson is that God's love is eternal and paramount. His anger does not remain forever.
For a brief moment I abandoned you,
but with great tenderness I will take you back. In an outburst of wrath, for a moment
I hid my face from you; But with enduring love I take pity on you,
says the Lord, your redeemer. (Isaiah 54:7-8)
Moses understood what Isaiah would later teach: any feelings of anger on God's part towards his people are momentary; his love and mercy are eternal.
ADDENDUM: Moses' attitude is perhaps better expressed in Dt. 9, referring to the same episode (after the Golden Calf), when he says:
Those forty days, then, and forty nights, I lay prostrate before the
Lord, because he had threatened to destroy you. 26 And I prayed to the
Lord and said: O Lord God, do not destroy your people, the heritage
you redeemed in your greatness and have brought out of Egypt with your
strong hand. 27 Remember your servants, Abraham, Isaac, and Jacob. Do
not look upon the stubbornness of this people nor upon their
wickedness and sin, 28 lest the land from which you have brought us
say, “The Lord was not able to bring them into the land he promised
them, and out of hatred for them, he brought them out to let them die
in the wilderness.” 29 They are your people and your heritage, whom
you have brought out by your great power and with your outstretched