No, Jesus with wouldn’t be advocating for homosexual people. But he also wouldn’t be protesting them. You are right, Jesus was and still remains a loving and just leader. But he always promotes his father’s law of what is approved of.
John 17:17 says: Sanctify them by means of the truth; your word is truth.”
Jesus believed and preached the word of his father Jehovah. He did it with love and care and unbiased. He’d go up to everyone and speak with them without making them feel bad about themselves no matter their background. However, he didn’t change or bend his message to people to fit whatever they thought was okay.
Luke 4:43 says: “But he said to them: “I must also declare the good news of the Kingodom of God to other Cities because for this I was sent.””
Jesus preached about God’s kingdom and the way into that kingdom has never and will never change. 1 Corinthians 6:9-11 says: “Or do you not know that unrighteous people will not inherit God’s Kingdom? Do not be misled. Those who are sexually immoral, idolaters, adulterers, men who submit to homosexual acts, men who practice homosexuality, thieves, greedy people, drunkards, revilers, and extortioners will not inherit God’s Kingdom. And yet that is what some of you were. But you have been washed clean; you have been sanctified; you have been declared righteous in the name of the Lord Jesus Christ and with the spirit of our God.”