In Matthew 23:23, Jesus said that faith is one of the weightier matters of the Law, so obedience to God's instructions is about expressing our faith in Him to guide us in how to rightly live. Living by faith is always associated with a willingness to obey God's instructions such as with every example of saving faith listed in Hebrews 11, whereas disobedience to God's instructions is referred to as breaking faith. In 1 John 5:3, to love God is the obey His commands, which are not burdensome, so it is about growing in a relationship with God based on faith and love.
"To fulfill the Law" means "to cause God's will (as made known in the Law) to be obeyed as it should be" (NAS Greek Lexicon 2c3). After Jesus said he came to fulfill the Law in Matthew 5, this is precisely what he then proceeded to do six times throughout the rest of the chapter by teaching how to correctly understand and obey it. In Galatians 5:14, loving your neighbor fulfills the entire law, so it refers to obeying the Law as it should be obeyed, and refers to something countless people have done, not to something unique to Christ. Likewise, Galatians 6:2 says that bearing one another's burdens fulfills the Law of Christ, which refers to obeying it as it should be obeyed, not to doing away with it. In Romans 15:18-19, it says that Paul fulfilled the Gospel, which again referred to causing Gentiles to become fully obedient to it in word and in deed, not to doing away with it.
In regard to Matthew 22:36-40, the reason why the greatest two commands are the greatest is because they summarize all of the other commands and the reason why all of the other commands hang on the greatest two is because they are examples of what it looks like to correctly obey them. The command to love the Lord our God with all of our heart, soul, mind, and strength is a lot easier said than done, so thankfully we have all of the other commands and Christ's example of obedience to those commands to paint us a picture of what that looks like. So love does not replace the Mosaic Law, but rather love is its essence. We can't obey God's command to love by disregarding all of His other instructions for how He wants us to love.
It is impossible to follow Jesus by refusing to follow the Law that he followed and taught his followers to follow by word and example. Following Jesus is not just for Jews and Gentiles can either choose to follow him or not, but there is no sense in claiming to follow him while refusing to follow him. While the Law was only given to Israel, it was never meant only for Israel because Israel was given the role by God to be a light to the nations, of blessing them by teaching them about Him, to turn from their wicked ways, and to walk in God's ways (Isaiah 2:2-3, Isaiah 49:6, Deuteronomy 4:5-8). There are many other verses that describe God's Law as being instructions for how to walk in His ways, such as Deuteronomy 10:12-13, Joshua 22:5, and Psalms 103:7, so it is not instructions for how to act like a Jew, but rather it is instructions to all of God's followers for how to reflect His attributes, such as holiness, righteousness, goodness (Romans 7:12), justice, mercy, faithfulness (Matthew 23:23), love, joy, peace, patience, kindness, gentleness, and self-control (Exodus 34:6-7, Galatians 5:21-22).
You can say that you don't prefer the death penalty for certain offences, but without being able to appeal to standard that is independent of human opinion, you have no way to establish that anyone has a moral obligation to do what you prefer instead of what they prefer, and vice versa. Morality is inherently a theistic concept.