The Pharisees thought that obedience and righteousness was motivated by their effort to keep the Mosiac law.
Under the New Covenant the Old Covenant of law and religion was abolished, Ephesians 2:15. Jesus nailed it to his cross, Colossians 2:14. The reason that it was abolished was because it was no longer applicable for New Testament Christians. Paul wrote, "Christ is the end of the law for righteousness" Romans 10:4.
Many Christians believe that the law is their teacher and their guide and that they should live by laws, rules and religion. This is exactly what the Pharisees thought. But the Bible teaches that the Holy Spirit is our teacher and our guide and not the law, John 16:13.
When God abolished the written law, he wrote the spiritual law on the hearts and minds of his people, Hebrews 8:10. The spiritual law is very different from the written law. The written law demands perfection or death. The spiritual law is an urging or prompting to do what is right. There is no condemnation for those that violate the spiritual law, Romans 8:1.
Under the New Covenant God's people don't live by rules, laws or religion. Paul wrote, "The just shall live by faith" Romans 1:17. They live by faith in Christ and his Gospel.
I think this is probably a misunderstanding of the purpose of the Torah, what obedience entails and why it's important. It's typical to see the commands in the torah as basically arbitrary rules to be followed to the letter. There seems to me to be a general thought among many Christians that the Torah's purpose was either to set Israel apart from the nations, demonstrate that humans are stuck in sin and can't follow it's commands, or bring about the Messiah (or perhaps some combination of all of those).
I think many times there's a hidden motive too. Many of the commands appear to be arcane, archaic, or extreme, and so some people are motivated to find reasons to not-follow the commands. Sabbath is the most obvious that comes up frequently in theological debate, but many other are viewed with suspicion too such as the punishments prescribed by some laws, Levirate marriages, and a host of other specifics commands.
(1) So my first point would be that the purpose of the torah should not be understood as an arbitrary collection of commands issue by divine decree for no reason other than to show that, like Israel, we're sinners in need of a messiah. Instead there is more going on here. The Torah itself describes it's purpose as the way to life (Deut 11:9, 22:7, 22:15. 28:66, 30:15-20) so that if you follow the torah, you'll have a long life, and if you don't follow it's commands, you won't have a long life (note, this is refering to life here on earth, not eternal life after death in heaven). To Israel, this is contained in a covenant/treaty form and attached to specific stipulations about inheriting land. If they obeyed, they'd be granted favor in the promised land, but if they disobeyed, they'd be expelled from the promised land.
But I think it also applies more generally - if you follow the commands in the torah, you'll likely have a longer and more fulfilling life - if you do the opposite of it's commands, and so reject it, you'll likely live a shorter and less fulfilling life. I think this generalization is appropriate because all of the nations in the ancient world at least stretching from the Eastern Med to Persia, had very similar law codes (in some cases almost identical) so that there is reason to believe that in the ancient near eastern world there was a common law tradition in play and that Torah should be considered within this context. Somehow, they had all generally come to realize that if this common law tradition was followed, they'd fair better, but if they rejected this common law tradition, they'd fair poorer. In Israel, this common law was written and contained within a covenant document with some stipulations that were specific to Israel, but the generality of the law is still applicable. Example: you don't have to observe commands to not-murder and not-steal, but if you do those things, you're probably going to have a shorter and more miserable life. So in some sense, yes, you do have to observe them if you want to live a fulfilling life.
In general, the entire purpose of the Torah, and it's ancient near eastern counterparts (eg, the Code of Hammurabi) and the Ancient Near Eastern common law tradition was ideally to promote human flourishing. Even in the Torah, this is the very first command given to man (be fruitful, multiply, fill the earth and rule it). Most, if not all, of it's commands can be understood within this framework. It does this by protecting and promoting things like life, religion, property, family, social order, and peace.
(2) My second point is that like its near eastern counterparts, the torah did not, in it's original setting and intent, function legislatively. It's more common law than statutory law. And this is why in both Israel and the ancient near east, you can find law codes with specific wording, but you can find actual judicial cases that don't follow laws to the letter. These laws codes, Torah included, probably initially functioned as specific case law, legal guidance, legal wisdom, and premier examples - again, common law. As such, its commands are not always exhaustive or comprehensive of every situation that could arise as a legal matter. As common law, neither judges, kings, priests, elders, etc... were obliged to follow to the letter without applying common sense, wisdom, consideration for specific circumstances, or the merit of specific cases (nor does this mean they could ignore law either).
So should we obey the law? Well, the OP prescribed a way of obedience in the way the Pharisees prescribed. I'm saying there is reason to say we should follow Torah, just not necessarily as the Pharisees prescribed. It seems one could understand the Torah's purpose, original intent and common law and arrive at the conclusion that, yes, we should follow Torah, but it doesn't have to be in the way it was understood and practiced by the Pharisees. You don't have to accept their axioms or method in order to keep and observe the law. Or as Paul might say, "the letter kills but the Spirit gives life" (2 Cor 3:6). There is more leeway in application of the Law than people might think.
For specific scholarly sources, I'd recommend Raymond Westbrook ("A History of Ancient Near Eastern Law"), Joshua Berman ("Ani Maamin"), or John Walton ("The Lost World of the Torah").