What is your opinion on applying the phrase "a sin that admits you to Paradise" to a sin that bequeaths its perpetrator brokenness, humility, and remorse, or to a good deed that bequeaths its perpetrator vanity, arrogance, and a sense of entitlement, based on the saying of some of the Salaf?

Entering Paradise is not because of sin, but because of repentance. Sin may be more beneficial to the servant if it is accompanied by repentance, as it instills humility, brokenness, repentance, seeking forgiveness, and regret, thereby becoming a cause for his salvation. Whereas righteous deeds may instill conceit and arrogance. This does not mean advocating for committing sins; rather, the intent is that if one sins and then regrets it with humility and brokenness, that will benefit him.