From my experience, few seem to sense the need to be saved and the urgency to escape the wrath of God. Why? And how can we give people this sense?

Or, for such people who have little conviction of sin, perhaps we should ignore God's justice and present Him as a sort of Santa Claus figure just focussing on His love and the fact the gift is freely available?

I seem to have run into many "Christians" who are products of such a philosophy. And "as they have received Christ Jesus the Lord, so they walk in him". They received him as a sort of Santa Claus figure without conviction of sin and the fear of God, and so they walk - doing whatever they want to do with no regards to submission to God.

Though righteousness cannot be obtain by a person's performance

"know that a man is not justified by observing the law, but by faith in Jesus Christ. So we, too, have put our faith in Christ Jesus that we may be justified by faith in Christ and not by observing the law, because by observing the law no one will be justified." Gal 2:16

because we all fall short of God's standards,

"For if a law had been given that could impart life, then righteousness would certainly have come by the law. But the Scripture declares that the whole world is a prisoner of sin, so that what was promised, being given through faith in Jesus Christ, might be given to those who believe." Gal 3:21,22
"for all have sinned and fall short of the glory of God" Rom 3:23

the law has a necessary purpose in leading us to Christ

"So the law was put in charge to lead us to Christ that we might be justified by faith." Gal 3:24

It does so by convicting us of sin

"Therefore no one will be declared righteous in his sight by observing the law; rather, through the law we become conscious of sin." Rom 3:20

This being the case, those who lack the conviction of sin or think that they can make it on their own are not qualified to receive eternal life

"You who are trying to be justified by law have been alienated from Christ; you have fallen away from grace." Gal 5:4
All who rely on observing the law are under a curse, for it is written: "Cursed is everyone who does not continue to do everything written in the Book of the Law." Gal 3:10

The fact that the law brings conviction of sin is the reason why it is presented first

What shall we say, then? Is the law sin? Certainly not! Indeed I would not have known what sin was except through the law. For I would not have known what coveting really was if the law had not said, "Do not covet." But sin, seizing the opportunity afforded by the commandment, produced in me every kind of covetous desire. For apart from law, sin is dead. Rom 7:7,8

An example from Acts:

Several days later Felix came with his wife Drusilla, who was a Jewess. He sent for Paul and listened to him as he spoke about faith in Christ Jesus. As Paul discoursed on righteousness, self-control and the judgment to come, Felix was afraid and said, "That's enough for now! You may leave. When I find it convenient, I will send for you." Acts 24:24,25
As Paul has also demonstrated in the book of Romans, he starts his presentation of the gospel with a discourse on sin and judgment with the hope that it would produce the fear of God's judgment and so drive one to consider the good news seriously.

This is why the religious leaders brought a woman caught in adultery to Jesus. They wanted to see if he agreed with the God of the Old Testament. And he did - He commanded that she be stoned to death - only that the person without sin be the first to throw a stone. And before he sent her away he commanded her to sin no more. (Little did the religious leaders know that they were speaking to the God of the Old Testament)

If people don't like God as He has presented Himself in the Old Testament,
then to hell with them (literally)!

If a person acknowledges that they are sinners, but they don't agree with what God calls sin or don't agree with His judgment against sin are they qualified to be saved? I think not! For how can you say that you believe in God, but don't agree with what He said. Isn't it like making the golden calf in the place of God worshipping it?

Yet often Christians seem to treat the Old Testament as if that's the way God used to be, but He's not like that anymore.

However, having said that, I do find the New Testament to contain sufficient and concise information on these matters to warrant starting a person off in it, rather than starting in the Old Testament. The synoptics gospels and the beginning of Romans particularly emphasize these points. But lets not skip or skim through them, for as zealous as some of us want to see those around us become believers, to do so without having firm convictions concerning sin and judgment often results in mere nominalism. And how can these be achieved? By presenting the gospel as God did throughout the Bible and as Jesus did, presenting issues of sin and judgment first.