What Is the Righteousness of the Pharisees That Ours Is Supposed to Exceed According to Jesus?

In what's called the Sermon on the Mount, Jesus says in Matthew 5:20:

For I say unto you, That except your righteousness shall exceed the righteousness of the scribes and Pharisees, ye shall in no case enter into the kingdom of heaven.

I've heard this explained in a number of different ways, often, I've found, in convenient ones to make room for false doctrine or practice.  One error I've heard says something like the following and maybe you've said it.  I'm going to indent it, so that you'll know it's representing what other people say it means:

The Pharisees were super righteous people.  They were fastidious at keeping the law, since they were experts and were so, so into the law.  They were very righteous people, just not perfect, which is what it had to be in order to be saved.

Furthermore, there are versions of Pharisees today.  They try to keep all the laws and are very strict.  This strictness is Pharisaical, and it produces people who are self-righteous and are trying to impress people with their righteousness by being stricter than others.

This representation of the "righteousness of the Pharisees" doesn't fit the context in the sermon of Jesus.  Jesus wasn't talking about how greatly righteous the Pharisees were, but how poor their righteousness was.  That is seen in the preceding and the proceeding context of Jesus' sermon.  I contend that evangelicals use this false interpretation of the sermon to attack both keeping the law and strict keeping of the law.

A misrepresentation of Jesus, that He wishes to disabuse His audience, was that He, as a teacher, was trying to destroy the law.  He says in verse 17:

Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil.

You could hear, "Just the opposite."  What Jesus came to preach didn't result in people not being righteous.  They couldn't and wouldn't be righteous the Pharisee way.  The Pharisees were the ones diminishing the law, not Jesus, and Jesus illustrates that in the post context of verses 21 to 48.  The standard remained God and not the Pharisees, as Jesus ends the chapter in verse 48.

As Jesus described His position on not destroying the law, He talked about the perpetuity of every jot and tittle (verse 18) and that the greatest in His kingdom kept the least of His commandments (verse 19).  The salvation that Jesus taught would produce righteous people.  They could and would keep the law -- more than that.

Jesus first illustrates His position by giving several illustrations of the application of "Thou shalt not kill."  His audience had been taught that particular law or standard of righteousness and if they were at the Pharisee level, they wouldn't still be keeping the law like Jesus taught that it should be kept.  Because of that, they weren't being righteous.

If Jesus' audience hated people in their heart, they were guilty of murder before God.  If they said certain hateful things, they were committing murder.  If they wouldn't reconcile with someone, they were as much murderers likewise.

Pharisaical righteousness was designed around something less than law keeping.  They didn't really keep the Sabbath, didn't really not murder, and didn't really not commit adultery.  They didn't really love God or their neighbor.

The Pharisees concocted means of appearing to keep the law or just keeping their own minimization of the law, what we might call today a deconstruction of the law.  With the Pharisees, you could keep the law without actually keeping it.  Jesus pointed this out again and again.

You don't have the righteousness of God when you have that of the Pharisees.  You weren't keeping the law, when you were a Pharisee.

There is an irony to the false interpretation.  It is Pharisaical.  It purposefully diminishes the law and therefore diminishes the righteousness of God.  What I'm saying also fits into what the Apostle Paul said that they did in Romans 10:3:

For they being ignorant of God's righteousness, and going about to establish their own righteousness, have not submitted themselves unto the righteousness of God.

The righteousness of justification by faith produces a righteousness greater than what the Pharisees believed and lived.  It would look like the righteousness of God, because it was a righteousness of the power of God.  This was having your house built on the rock of Jesus Christ and not the sand of the Pharisees.