why the Mike Yanagita scene is critical.

I've read analyses of that scene and figuring out the purpose doesn't make it any better.

It's a stupid setup for what happens later. The movie never, ever establishes that Marge was so unworldly she would take anyone at their word just because they were acting a certain way. As if that weren't bad enough, before that scene Marge is so unfazed when she sees the bodies of three murdered people that she doesn't even regard it with any shock or confusion. This implies that she's a seasoned veteran who's "seen it all" and therefore hardly the naive type.

TxMike - I watched this great film last night and thought about this scene.

I think you are right but there is a slight nuance here others here seem to be missing.

Marge knew right away that Jerry was lying or hiding something during their first meeting. You don't need to be an FBI profiler to pick up on Jerry's obvious body language and evasiveness (great performance by Macy second only to McDormand!).

After the first interview with Jerry, I believe Marge had made the most logical deduction : that Jerry KNEW a car had been stolen from his lot or taken by Shep and was covering it up because if it was discovered he could lose his job. Jerry was lying not because he was involved with the 3 murders she was investigating, but because he was afraid of being fired.

There was no reason for Marge to assume the sales manager at a car dealership would be connected to a triple homicide.

After Marge met with Mike and found out that virtually everything Mike told her was a lie, I think she realized that she really should go back and make sure Jerry was not connected to the murders. The way to do that would be to force him to admit that a car had been taken or stolen from his lot and that that was the only thing that connected Jerry to her case.

I saw this scene as being important for the very same reason. She realized she was being way too trusting of people. Here she had an old friend from school, who she knew for years, lie right to her face. So why should she believe a complete stranger? Thats the question she must've asked herself before going back to the car dealership.

Another thought I had while watching this for the first time is that Marge had not yet "incorporated her shadow" into her personality. Its something Jordan Peterson has spoke about, an idea first postulated by Carl Yung, that you have to incorporate the dark part of yourself to be a fully functioning capable person. That hiding your dark feelings and thoughts, burying them, keeps you from your full potential. As a police officer, Marge needed to be able to think the way evil people do in order to fight them. You have to be able to understand how they think so you can catch them. She was a sweet, trusting, and lovable person and she just didnt understand how people can lie and be so brutal. Even though she cracked the case when she stopped being too trusting, at the very end when she is taking Gaear Grimsrud in the back of her police car she still questions the whole crime. "Theres more to life than a little money, why would you people do all this". So she is still trying to come to terms with this evil act because she has a hard time thinking like them.

I guarantee after the whole event, she became a better police officer for the future.

It's just character development to show she is very mild mannered and liked in town. She's supposed to be something of a friendly but simple small town cop. This scene lets us know she's always lived here and is regarded as just one of the townsfolk, not a supercop or anything.

Presumably if she had not encountered Mike she would not have suspected Jerry was lying to her face and the case might not have been solved.

1. Jerry was not a suspect at that present time. No reason to put him through a big interrogation.
2. She only wanted to ask him a single question, nothing more.
3. She was the conversational and chatty type through the film. Him being evasive by changing the subject and therefore indirectly answering the question was a good strategy to say the least.

No reason to assume he's lying. I myself would have been a little suspicious at how evasive he was being, but I probably wouldn't put him on the spot right then and there. She was simply letting her cool-headed personality get to her a bit too much.