I don't understand your reasoning or why it would be so.
We should agree that "trying to save" and "wanting to save" are different things in language. If we don't use the language the same way there is no possibility to have a discussion. Or we have to start with discussing what you mean by "trying" and "wanting".
What was your point? I showed you your claim was wrong: "This means that everyone else wasn’t chosen." Of course God had a chosen people, and obviously only Israel was God's chosen people. But that was not only concerning salvation in the sense that only Israel would be saved and no other people. For an example what about the people of Nineveh?
The same as with the Canaanite woman.
She was not of the chosen people Israel Jesus came to save. Yet he healed her anyway.
But He answered and said, “I was sent only to the lost sheep of the house of Israel.” But she came and began to bow down before Him, saying, “Lord, help me!” And He answered and said, “It is not good to take the children’s bread and throw it to the dogs.” But she said, “Yes, Lord; but even the dogs feed on the crumbs which fall from their masters’ table.” Then Jesus said to her, “O woman, your faith is great; it shall be done for you as you wish.” And her daughter was healed at once.
— Matthew 15:24-28
No, you have given a possible understanding of
Titus 2:1-12 which doesn't fit
1 Timothy 2:1-6 very well. Most scholars also disagree with you